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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Polar coordinates</h3>
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<h2 class="hd hd-2 unit-title">2. Objectives</h2>
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<p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Convert between Cartesian and <b class="bf"><span style="color:#27408C">polar coordinates</span></b>. </p></li><li><p>
Set up integrals to <b class="bf"><span style="color:#27408C">compute area bounded by polar curves</span></b>. </p></li><li><p><b class="bf"><span style="color:#27408C">Graph</span></b> functions of the form [mathjaxinline]\, r=r(\theta )[/mathjaxinline]. </p></li></ul><p><b class="bfseries">Contents: 18 pages</b></p><p>
15 videos (88 minutes 1x speed) 38 questions </p>
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<h2 class="hd hd-2 unit-title">3. Review</h2>
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<p>
Find the [mathjaxinline]\, x[/mathjaxinline]- and [mathjaxinline]\, y[/mathjaxinline]-coordinates of the point.<br/>(You can enter your numerical answer in functional form, e.g. [mathjaxinline]\tan (\pi /3)[/mathjaxinline].)<br/></p>
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<p style="display:inline">[mathjaxinline]x =\,[/mathjaxinline]</p>
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<div class="caption"><b>Figure 5</b>: <span>The point with [mathjaxinline]\, x=3,\, y=4[/mathjaxinline].</span></div>
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Find the length [mathjaxinline]\, r\,[/mathjaxinline] and the angle [mathjaxinline]\, \theta \,[/mathjaxinline].<br/>(You can enter your numerical answer in functional form, e.g. [mathjaxinline]\tan (\pi /3)[/mathjaxinline].)<br/><p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_polar-tab3-problem2_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<h2 class="hd hd-2 unit-title">4. Introduction to polar coordinates</h2>
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The <span style="color:#27408C"><b class="bf">polar coordinates</b></span> of a point [mathjaxinline]\, P\,[/mathjaxinline] are an ordered pair [mathjaxinline]\, (r,\theta )\,[/mathjaxinline] such that </p><table id="a0000000271" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000272"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =r\cos (\theta )[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.165)</td></tr><tr id="a0000000273"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y[/mathjaxinline]
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[mathjaxinline]\displaystyle =r\sin (\theta ),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.166)</td></tr></table><p>
where the ordered pair [mathjaxinline]\, (x, y)\,[/mathjaxinline] give the rectangular coordinates of the point [mathjaxinline]P[/mathjaxinline].<br/></p><p>
In other words, given the polar coordinates [mathjaxinline]\, (r,\theta )\,[/mathjaxinline] of a point, we can find its [mathjaxinline]\, x[/mathjaxinline]- and [mathjaxinline]\, y[/mathjaxinline]-coordinates using these formulas. <br/></p><p>
The usual rectangular coordinates [mathjaxinline]\, x\,[/mathjaxinline] and [mathjaxinline]\, y\,[/mathjaxinline] are also called <span style="color:#27408C"><b class="bf">Cartesian coordinates</b></span>. To find polar coordinates from Cartesian coordinates, we use </p><table id="a0000000274" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000275"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle r[/mathjaxinline]
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[mathjaxinline]\displaystyle =\pm \sqrt {x^2+y^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.167)</td></tr><tr id="a0000000276"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \theta[/mathjaxinline]
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[mathjaxinline]\displaystyle =\, \arctan \left(\frac{y}{x}\right).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.168)</td></tr></table><p>
However, [mathjaxinline]\, r\,[/mathjaxinline] and [mathjaxinline]\, \theta \,[/mathjaxinline] are not unique for any given point. We will explain this ambiguity along with how to use these formulas in more detail after a few exercises.<br/></p><p>
Polar coordinates are motivated by the fact that we can locate a point on a plane by specifying: </p><table id="a0000000277" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000278"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle r:[/mathjaxinline]
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[mathjaxinline]\displaystyle \text {the distance from the origin to the point,}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.169)</td></tr><tr id="a0000000279"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \theta :[/mathjaxinline]
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[mathjaxinline]\displaystyle \text {the angle of the ray from the origin to the point with the positive}\, x\text {-axis}.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.170)</td></tr></table>
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Find the polar coordinates, with [mathjaxinline]\, r\geq 0[/mathjaxinline] and [mathjaxinline]\, 0\leq \theta &lt; 2\pi ,\,[/mathjaxinline] of the following points. </p>
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[mathjaxinline]P_1:\, (x,y)\, =\, (4,0)[/mathjaxinline]</td>
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<p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta \, =\,[/mathjaxinline]</p>
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[mathjaxinline]P_2:\, (x,y)\, =\, (0,-8)[/mathjaxinline]</td>
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<p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta \, =\,[/mathjaxinline]</p>
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[mathjaxinline]P_3:\, (x,y)\, =\, (-2,0)[/mathjaxinline]</td>
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<p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta \, =\,[/mathjaxinline]</p>
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[mathjaxinline]P_4:\, (x,y)\, =\, (4,4)[/mathjaxinline]</td>
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<p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta \, =\,[/mathjaxinline]</p>
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[mathjaxinline]P_5:\, (x,y)\, =\, (-6,-6\sqrt {3})[/mathjaxinline]&#8195;</td>
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<p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta \, =\,[/mathjaxinline]</p>
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Convert to rectangular coordinates
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Find the Cartesian coordinates of the following using the formulas </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000289" style="table-layout:auto" width="100%">
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =r\cos (\theta )[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.176)</td>
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle y[/mathjaxinline]
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[mathjaxinline]\displaystyle =r\sin (\theta )[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.177)</td>
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even when [mathjaxinline]\, r&lt;0[/mathjaxinline].<br/></p>
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[mathjaxinline]\displaystyle P_1\, :\, (r_1,\theta _1)\, =\, \left(2,-\frac{\pi }{3}\right):\quad[/mathjaxinline]</td>
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[mathjaxinline]\displaystyle P_2\, :\, (r_2,\theta _2)\, =\, \left(-4, \frac{7\pi }{6}\right):\quad[/mathjaxinline]</td>
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<p>
The polar coordinates describing a point are not unique. First, </p><table id="a0000000297" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000298"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle (r,\theta )[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(r, \theta + 2\pi n\right) \qquad (n\, \text {any integer}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.182)</td></tr></table><p>
That is , knowing the [mathjaxinline]x[/mathjaxinline]- and [mathjaxinline]y[/mathjaxinline]- coordinates only determines [mathjaxinline]\theta[/mathjaxinline] up to [mathjaxinline]2\pi[/mathjaxinline]-periodicity. We frequently use conventions such as: </p><table id="a0000000299" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000300"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 0\leq \theta < 2\pi[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.183)</td></tr><tr id="a0000000301"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {or}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle -\pi < \theta \leq \pi .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.184)</td></tr></table><p>
Second, </p><table id="a0000000302" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000303"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
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[mathjaxinline]\displaystyle (-r,\theta )[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(r, \theta \pm \pi \right) \qquad -\infty <r<\infty[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.185)</td></tr><tr id="a0000000304"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {and equivalently}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle (r,\theta )[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(-r, \theta \pm \pi \right) \qquad -\infty <r<\infty .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.186)</td></tr></table><p>
Verify that the first relation (and therefore the second) above makes sense since the left hand side and the right hand side indeed give the same [mathjaxinline]x[/mathjaxinline]- and [mathjaxinline]y[/mathjaxinline]-coordinates: </p><table id="a0000000305" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000306"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle x[/mathjaxinline]
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[mathjaxinline]\displaystyle = (-r) \cos \left(\theta \right)[/mathjaxinline]
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[mathjaxinline]\displaystyle = r \cos \left(\theta \pm \pi \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.187)</td></tr><tr id="a0000000307"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle = (-r) \sin \left(\theta \right)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =r \sin \left(\theta \pm \pi \right).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.188)</td></tr></table>
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Equivalent polar coordinates
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Which of the following polar coordinates are equivalent to [mathjaxinline]\, \displaystyle (r,\theta )=\left(5,\frac{17\pi }{5}\right)[/mathjaxinline]?<br/>(Check all that apply.)<br/></p>
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<text>[mathjaxinline]\displaystyle \left(5,-\frac{7\pi }{5}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left(-5,\frac{7\pi }{5}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left(-5,\frac{12\pi }{5}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left(-5,-\frac{2\pi }{5}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left(5,\frac{7\pi }{5}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left(5,\frac{-3\pi }{5}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left(5,\frac{12\pi }{5}\right)[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle \left(-5,\frac{2\pi }{5}\right)[/mathjaxinline]</text>
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Plot the following on the graph below: </p>
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Point 1: [mathjaxinline]\displaystyle \, (r_1,\theta _1)\, =\, \left(2, \frac{11\pi }{6}\right)[/mathjaxinline]<br/></p>
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Point 2:[mathjaxinline]\displaystyle \, (r_2,\theta _2)\, =\, \left(-\sqrt {2}, \frac{\pi }{4}\right)[/mathjaxinline]<br/></p>
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Point 3:[mathjaxinline]\displaystyle \, (r_3,\theta _3)\, =\, \left(-2, -\frac{\pi }{3}\right)[/mathjaxinline]<br/></p>
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<h2 class="hd hd-2 unit-title">6. The angle theta</h2>
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Note the following procedure for finding [mathjaxinline]\theta[/mathjaxinline] works for finding [mathjaxinline]\theta[/mathjaxinline] between [mathjaxinline]0[/mathjaxinline] and [mathjaxinline]2\pi[/mathjaxinline]. You may need to modify as needed to find [mathjaxinline]\theta[/mathjaxinline] between [mathjaxinline]-\pi[/mathjaxinline] and [mathjaxinline]\pi[/mathjaxinline] for example! </p>
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<p>
To find [mathjaxinline]\, \theta ,[/mathjaxinline] we first find </p><table id="a0000000324" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000325"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \theta _0[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\arctan \left(\frac{|y|}{|x|}\right).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.200)</td></tr></table><div id="a0000000326" class="figure"><center><img src="/assets/courseware/v1/2c8c43872a73b75e26d26e284a322dbf/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_theta0.svg" width="350px" alt="See caption" style="margin: 10px 25px 25px 25px"/><div class="caption"><b>Figure 8</b>: <span>Here, [mathjaxinline](x,y)\,[/mathjaxinline] lies in the second quadrant, [mathjaxinline]\, \theta =\pi -\theta _0\,[/mathjaxinline].</span></div><br/></center></div><p>
Then we find [mathjaxinline]\, \theta \,[/mathjaxinline] using [mathjaxinline]\, \theta _0,\,[/mathjaxinline] by considering which quadrant it lies in, which is best done using a picture like the one above. </p>
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Convert to polar coordinates: with computation
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Find the polar coordinates of the following using the convention [mathjaxinline]\, r\geq 0[/mathjaxinline] and [mathjaxinline]\, -\pi &lt; \theta \leq \pi[/mathjaxinline].<br/></p>
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(You can enter your answer as math expression, e.g. [mathjaxinline]\arctan {3}[/mathjaxinline].)<br/></p>
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Point 1:</td>
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[mathjaxinline](x,y)\, =\, (-4,3)[/mathjaxinline]&#8195;</td>
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<p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta \, =\,[/mathjaxinline]</p>
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Point 2:</td>
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[mathjaxinline](x,y)\, =\, (3,-2)[/mathjaxinline]&#8195;</td>
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<p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta \, =\,[/mathjaxinline]</p>
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<p>
An equation written in polar coordinates is called a <span style="color:#27408C"><b class="bf">polar equation</b></span>. We will mostly be dealing with polar equations of the form [mathjaxinline]\, r=r(\theta )[/mathjaxinline]. </p><p>
The simplest examples are: </p><ul class="itemize"><li><p>
[mathjaxinline]r=a[/mathjaxinline]: </p><div class="figure"><center><img src="/assets/courseware/v1/60360aea8787841c7da914903200a10c/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_circle.svg" width="350px" alt="Polar circle" style="margin: 10px 25px 25px 25px"/><br/></center></div></li><li><p>
[mathjaxinline]\theta = \alpha[/mathjaxinline] ([mathjaxinline]r\geq 0[/mathjaxinline]):<br/></p><div class="figure"><center><img src="/assets/courseware/v1/63b0357df06f8bb36a4b3f39ea3c9abe/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_ray.svg" width="350px" alt="Polar ray" style="margin: 10px 25px 25px 25px"/><br/></center></div><p>
If we use the convention [mathjaxinline]-\infty <r<\infty ,\,[/mathjaxinline] then [mathjaxinline]\, \theta =\alpha \,[/mathjaxinline] is a line through the origin: </p><div class="figure"><center><img src="/assets/courseware/v1/3c75c1c037502bbe88e74bb3cc488854/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_line.svg" width="350px" alt="Polar line" style="margin: 10px 25px 25px 25px"/><br/></center></div></li></ul><p>
We use the circles and rays as the grid of the polar coordinate systems. </p>
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Sketch the graph of the polar equation [mathjaxinline]r=-2[/mathjaxinline] for [mathjaxinline]\, 0\leq \theta \leq \pi /2[/mathjaxinline].<br/></p>
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<h2 class="hd hd-2 unit-title">8. Lines in polar coordinates</h2>
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Different domain
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Consider the same polar equation as in the video but with a different range of [mathjaxinline]\theta[/mathjaxinline] values: </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000341" style="table-layout:auto" width="100%">
<tr id="a0000000342">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle r=\frac{1}{\sin (\theta )}\qquad \text {for}\, \, \, \pi &lt;\theta &lt;2\pi .[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.211)</td>
</tr>
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Plot the two points </p>
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<p>
Point 1: [mathjaxinline]\, \, \, (r(\theta ),\, \theta )\,[/mathjaxinline] at [mathjaxinline]\, \displaystyle \theta =\frac{3\pi }{2},\,[/mathjaxinline] </p>
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Point 2: [mathjaxinline]\, \, \, (r(\theta ),\, \theta )\,[/mathjaxinline] at [mathjaxinline]\, \displaystyle \theta =\frac{7\pi }{4},\,[/mathjaxinline] </p>
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and sketch the polar curve given by the equation and domain above.<br/></p>
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Vertical line: equation
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Find the polar equation [mathjaxinline]\, r=r(\theta )\,[/mathjaxinline] for the vertical line that passes through the [mathjaxinline]x[/mathjaxinline]-axis at [mathjaxinline]2[/mathjaxinline].<br/>(Enter [mathjaxinline]\, r[/mathjaxinline] in terms of [mathjaxinline]\, \theta[/mathjaxinline].)<br/></p>
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<p style="display:inline">[mathjaxinline]r \, =\,[/mathjaxinline]</p>
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Vertical line: domain
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Consider the polar equation [mathjaxinline]\, r=r(\theta )\,[/mathjaxinline] you obtained in the previous problem for the line [mathjaxinline]x=2[/mathjaxinline] .<br/></p>
<p>
Find [mathjaxinline]\, \theta _1\, &lt; \, \theta _2\,[/mathjaxinline] in the interval [mathjaxinline]\, [-\pi ,\, \pi \, ]\,[/mathjaxinline] such that </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000353" style="table-layout:auto" width="100%">
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle r(\theta )\geq 0\quad \text { for all} \, \, \, \theta _1 &lt;\theta &lt;\theta _2,[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.218)</td>
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and as [mathjaxinline]\, \theta \,[/mathjaxinline] increases from [mathjaxinline]\, \theta _1\,[/mathjaxinline] to [mathjaxinline]\, \theta _2,\,[/mathjaxinline] the point [mathjaxinline](r(\theta ), \theta )\,[/mathjaxinline] traces the entire vertical line. <br/></p>
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<p style="display:inline">[mathjaxinline]\theta _1 \, =\,[/mathjaxinline]</p>
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Generic line: equation
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Find the polar equation [mathjaxinline]\, r=r(\theta )\,[/mathjaxinline] for [mathjaxinline]\, y=mx+b\,[/mathjaxinline] by first writing [mathjaxinline]\, x,y\,[/mathjaxinline] in terms of [mathjaxinline]\, r,\, \theta[/mathjaxinline] and then isolating [mathjaxinline]r[/mathjaxinline].<br/></p>
<p>
(Enter your answer in terms of the polar variables [mathjaxinline]\, r\,[/mathjaxinline], [mathjaxinline]\, \theta \,[/mathjaxinline], and the parameters [mathjaxinline]\, b\,[/mathjaxinline] and [mathjaxinline]\, m[/mathjaxinline].)<br/></p>
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<p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p>
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</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h3 class="hd hd-3 problem-header" id="polar-tab8-problem5-problem-title" aria-describedby="block-v1:MITx+18.01.3x+1T2020+type@problem+block@polar-tab8-problem5-problem-progress" tabindex="-1">
Generic Line: domain
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<p>
Let [mathjaxinline]r=r(\theta )\,[/mathjaxinline] be the polar equation you obtain for the line [mathjaxinline]y=mx+b[/mathjaxinline] in the previous problem.<br/></p>
<p>
Assume [mathjaxinline]\, m ,b&gt;0[/mathjaxinline].<br/></p>
<p>
Find [mathjaxinline]\, \theta _1\, &lt;\, \theta _2\,[/mathjaxinline] in the interval [mathjaxinline]\, [0, 2\pi ]\,[/mathjaxinline] such that </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000361" style="table-layout:auto" width="100%">
<tr id="a0000000362">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle r(\theta )\geq 0\quad \text { for all}\, \, \, \, \theta _1 &lt;\theta &lt;\theta _2[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.223)</td>
</tr>
</table>
<p>
and as [mathjaxinline]\, \theta \,[/mathjaxinline] increases from [mathjaxinline]\, \theta _1\,[/mathjaxinline] to [mathjaxinline]\, \theta _2,\,[/mathjaxinline] the point [mathjaxinline](r(\theta ), \theta )\,[/mathjaxinline] traces the entire line [mathjaxinline]y=mx+b\,[/mathjaxinline].<br/></p>
<p>
(Enter [mathjaxinline]\, \theta _1\,[/mathjaxinline] and [mathjaxinline]\, \theta _2\,[/mathjaxinline] as a function of [mathjaxinline]\, m\,[/mathjaxinline] and/or [mathjaxinline]\, b[/mathjaxinline].)<br/></p>
<p>
<p style="display:inline">[mathjaxinline]\theta _1 \, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_polar-tab8-problem5_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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\(\)
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<p style="display:inline">[mathjaxinline]\theta _2 \, =\,[/mathjaxinline]</p>
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\(\)
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
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<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h3 class="hd hd-3 problem-header" id="polar-tab8-problem6-problem-title" aria-describedby="block-v1:MITx+18.01.3x+1T2020+type@problem+block@polar-tab8-problem6-problem-progress" tabindex="-1">
Review: Line passing through the origin
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<p>
Use the convention [mathjaxinline]-\infty &lt;r&lt;\infty[/mathjaxinline]. </p>
<p>
Which of the following are equations for the line [mathjaxinline]\, y=mx\,[/mathjaxinline] for [mathjaxinline]\, m&lt;0[/mathjaxinline]?<br/>(Check all that apply.)<br/></p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_polar-tab8-problem6_2_1">
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<text>[mathjaxinline]\theta \, =\, -\arctan (m)[/mathjaxinline]</text>
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<text>[mathjaxinline]\theta \, =\arctan (-m)[/mathjaxinline]</text>
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<text>[mathjaxinline]\theta \, =\, \arctan (m)[/mathjaxinline]</text>
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<text>[mathjaxinline]\theta \, =\arctan (m)+\pi[/mathjaxinline]</text>
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<text>[mathjaxinline]\theta \, =\arctan (m)-\pi[/mathjaxinline]</text>
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<text>[mathjaxinline]\displaystyle r=\frac{1}{\sin \left(\theta -\arctan (m)\right)}[/mathjaxinline]</text>
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Circle with center on the y-axis: equation
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Find the polar equation [mathjaxinline]\, r=r(\theta )\,[/mathjaxinline] for the graph of [mathjaxinline]x^2+(y-b)^2=b^2[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p>
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Circle with center on the y-axis: domain
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As above, consider the circle [mathjaxinline]\, x^2+(y-b)^2=b^2[/mathjaxinline], for [mathjaxinline]b&gt;0[/mathjaxinline].<br/></p>
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<img alt="See text above" src="/assets/courseware/v1/997d0cd83f335236973f2324ddacd469/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_circleonyaxis.svg" style="margin: 10px 25px 25px 25px" width="350px"/>
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Find [mathjaxinline]\, \theta _1,\, &lt;\, \theta _2\,[/mathjaxinline] in [mathjaxinline]\, [0,2\pi ]\,[/mathjaxinline] such that </p>
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<p>
[mathjaxinline]\displaystyle r\left(\theta _1\right)= r\left(\theta _2\right)=0[/mathjaxinline];<br/></p>
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<p>
[mathjaxinline]\, r\geq 0\,[/mathjaxinline] for all [mathjaxinline]\, \theta _1 \leq \theta \leq \theta _2[/mathjaxinline];<br/></p>
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<p>
and as [mathjaxinline]\, \theta \,[/mathjaxinline] increases from [mathjaxinline]\, \theta _1\,[/mathjaxinline] to [mathjaxinline]\, \theta _2,\,[/mathjaxinline] the point [mathjaxinline]\, \left(r(\theta ), \theta \right)\,[/mathjaxinline] traces the circle <b class="bf">once</b>.</p>
<p>
<p style="display:inline">[mathjaxinline]\theta _1 \, =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta _2 \, =\,[/mathjaxinline]</p>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
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<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h3 class="hd hd-3 problem-header" id="polar-tab9-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.3x+1T2020+type@problem+block@polar-tab9-problem3-problem-progress" tabindex="-1">
When r is negative
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<p>
As above, consider the circle [mathjaxinline]\, x^2+(y-b)^2=b^2.[/mathjaxinline]<br/></p>
<div class="figure">
<center>
<img alt="See text above" src="/assets/courseware/v1/997d0cd83f335236973f2324ddacd469/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_circleonyaxis.svg" style="margin: 10px 25px 25px 25px" width="350px"/>
<br/>
</center>
</div>
<p>
This time, allow [mathjaxinline]-\infty &lt;r&lt;\infty[/mathjaxinline].<br/></p>
<p>
Find [mathjaxinline]\, \theta _1,\, &lt;\, \theta _2[/mathjaxinline] in [mathjaxinline]\, [0,2\pi ]\,[/mathjaxinline] such that </p>
<ul class="itemize">
<li>
<p>
[mathjaxinline]\, r(\theta )\leq 0[/mathjaxinline] for all [mathjaxinline]\, \theta _1 \leq \theta \leq \theta _2[/mathjaxinline];<br/></p>
</li>
<li>
<p>
[mathjaxinline]\displaystyle r\left(\theta _1\right)= r\left(\theta _2\right)=0[/mathjaxinline];<br/></p>
</li>
</ul>
<p>
and as [mathjaxinline]\, \theta \,[/mathjaxinline] increases from [mathjaxinline]\, \theta _1\,[/mathjaxinline] to [mathjaxinline]\, \theta _2,\,[/mathjaxinline] the point with polar coordinate [mathjaxinline]\, (r, \theta )\,[/mathjaxinline] traces the circle <b class="bf">once</b> in the counter-clockwise direction. </p>
<p>
<p style="display:inline">[mathjaxinline]\theta _1 \, =\,[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_polar-tab9-problem3_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<p style="display:inline">[mathjaxinline]\theta _2 \, =\,[/mathjaxinline]</p>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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</tr>
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<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Shifting theta
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In this problem, we write the polar equation for the circle in the previous problem, which is centered on the [mathjaxinline]y[/mathjaxinline]-axis, by shifting [mathjaxinline]\, \theta \,[/mathjaxinline] in the equation of a circle centered on the [mathjaxinline]x[/mathjaxinline]-axis.<br/></p>
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<p>
Recall from the video that the polar equation of the (orange) circle with radius [mathjaxinline]\, b\,[/mathjaxinline] centered at [mathjaxinline]\, (x,y)\, =\, (b,0)\,[/mathjaxinline] is<br/></p>
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<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle r[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 2b\cos (\theta ),\qquad \, -\frac{\pi }{2}\leq \theta \leq \frac{\pi }{2}.[/mathjaxinline]
</td>
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.235)</td>
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<p>
Find the angle [mathjaxinline]\, 0\leq \alpha &lt;2\pi \,[/mathjaxinline] so that the equation of the (blue) circle with radius [mathjaxinline]\, b\,[/mathjaxinline] centered at [mathjaxinline]\, (x,y)\, =\, (0,b),\,[/mathjaxinline] is </p>
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[mathjaxinline]\displaystyle \displaystyle r[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle 2b\cos (\theta -\alpha ),\qquad \, -\frac{\pi }{2}+\alpha \, \leq \, \theta \, \leq \, \frac{\pi }{2}+\alpha .[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.236)</td>
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<p style="display:inline">[mathjaxinline]\alpha =\,[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">10. Rotation about the origin</h2>
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The graph of [mathjaxinline]\, r=r(\theta -\alpha )\,[/mathjaxinline] is obtained by rotating the graph of [mathjaxinline]\, r=r(\theta )\,[/mathjaxinline] <b class="bf">about the origin</b> by the angle [mathjaxinline]\, +\alpha[/mathjaxinline].<br/></p><p>
If [mathjaxinline]\, \alpha >0\,[/mathjaxinline] the rotation is counterclockwise.<br/></p><p>
if [mathjaxinline]\, \alpha <0\,[/mathjaxinline] the rotation is clockwise.<br/></p>
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A circle touching the origin
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<div class="caption"><b>Figure 14</b>: <span> A circle with radius [mathjaxinline]\, 3\,[/mathjaxinline] that touches the origin and is centered on the ray [mathjaxinline]\, \displaystyle \theta =\frac{5\pi }{7}.\,[/mathjaxinline] <br/></span></div>
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Find a polar equation [mathjaxinline]\, r=r(\theta )\, \,[/mathjaxinline] along with [mathjaxinline]\, \theta _1\,[/mathjaxinline] and [mathjaxinline]\, \theta _2\,[/mathjaxinline] in [mathjaxinline]\, [0,2\pi ]\,[/mathjaxinline] such that </p>
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \displaystyle r(\theta )\geq 0\, \quad \text {for all}\, \, \theta _1\leq \theta \leq \theta _2,\,[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.240)</td>
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and as [mathjaxinline]\, \theta \,[/mathjaxinline] increases from [mathjaxinline]\, \theta _1\,[/mathjaxinline] to [mathjaxinline]\, \theta _2,\,[/mathjaxinline] the point with polar coordinate [mathjaxinline]\, (r, \theta )\,[/mathjaxinline] traces the circle shown in the diagram above <b class="bf">once</b> in the counter-clockwise direction.<br/></p>
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<p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta _1 =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta _2 =\,[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h3 class="hd hd-3 problem-header" id="polar-tab10-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.3x+1T2020+type@problem+block@polar-tab10-problem2-problem-progress" tabindex="-1">
Line
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Consider the line whose shortest distance to the origin is [mathjaxinline]5[/mathjaxinline] and that is <b class="bf">perpendicular</b> to the ray [mathjaxinline]\, \displaystyle \theta =\frac{5\pi }{7}\,[/mathjaxinline] for [mathjaxinline]\, r&gt;0[/mathjaxinline].<br/></p>
<p>
Find its polar equation [mathjaxinline]\, r=r(\theta ),\, \,[/mathjaxinline] and [mathjaxinline]\, \theta _1\, &lt; \, \theta _2\,[/mathjaxinline] in the interval [mathjaxinline]\, [0,2\pi ]\,[/mathjaxinline] such that [mathjaxinline]\, r(\theta )\geq 0\,[/mathjaxinline] for all [mathjaxinline]\, \theta _1\leq \theta \leq \theta _2,\,[/mathjaxinline] and as [mathjaxinline]\, \theta \,[/mathjaxinline] increases from [mathjaxinline]\, \theta _1\,[/mathjaxinline] to [mathjaxinline]\, \theta _2,\,[/mathjaxinline] the point [mathjaxinline]\, (r(\theta ),\, \theta )\,[/mathjaxinline] traces the entire line <b class="bf">once</b>.<br/></p>
<p>
<p style="display:inline">[mathjaxinline]r =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta _1 =\,[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\theta _2 =\,[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">11. Area element</h2>
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Review: Area of a piece of pizza
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Find the area [mathjaxinline]\, A[/mathjaxinline]&#8201; of a slice of pizza with radius [mathjaxinline]a[/mathjaxinline] (inches) and spanned by the angle [mathjaxinline]\, \theta ,\,[/mathjaxinline] as shown in the figure.<br/></p>
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<p style="display:inline">[mathjaxinline]A\, =\,[/mathjaxinline]</p>
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[mathjaxinline]\displaystyle \displaystyle \Delta A[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \approx \frac{1}{2} r^2 \Delta \theta[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table></span></div></center></div><p>
Passing to the differential, the area element [mathjaxinline]\, dA\,[/mathjaxinline] in polar coordinates is the area of the infinitesimal region spanned by [mathjaxinline]d\theta[/mathjaxinline], given by </p><table id="a0000000414" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000415"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle d A[/mathjaxinline]
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[mathjaxinline]\displaystyle = \frac{1}{2} r(\theta )^2 d\theta .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.250)</td></tr></table><div class="figure"><center><img src="/assets/courseware/v1/f4aa4b23083cc1c75faa743177fbc0cb/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_areaIntegral.svg" width="400px" alt="" style="margin: 10px 25px 25px 25px"/></center></div><p>
The area bounded a curve [mathjaxinline]r=r(\theta )[/mathjaxinline], and the two rays [mathjaxinline]\, \theta =\theta _1\,[/mathjaxinline] and [mathjaxinline]\, \theta =\theta _2\,[/mathjaxinline] is </p><table id="a0000000416" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000417"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle A[/mathjaxinline]
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[mathjaxinline]\displaystyle =\int _{\theta _1}^{\theta _2} d A[/mathjaxinline]
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[mathjaxinline]\displaystyle = \frac{1}{2}\int _{\theta _1}^{\theta _2} r(\theta )^2 d\theta[/mathjaxinline]
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<span class="trailing_text" id="trailing_text_polar-tab11-problem2_2_2"> [mathjaxinline] d\theta [/mathjaxinline]</span>
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Evaluate the integral and verify your answer by computing the area of the shaded triangle directly. <br/>(Enter your answer in terms of [mathjaxinline]\, a,\,[/mathjaxinline] and [mathjaxinline]\, b\,[/mathjaxinline] in the box below.)<br/></p>
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<p style="display:inline">Area [mathjaxinline]\, =\quad[/mathjaxinline]</p>
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<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
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<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
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<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
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<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
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<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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Area of part of the disk
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<p>
Consider the same circle as in the video.<br/></p>
<div class="figure">
<center>
<img alt="Circle of radius a centered on x axis at a. Region bounded by a line of positive angle alpha and the top left porition of circle." src="/assets/courseware/v1/69f5c75936c2bfc9004d12b15bd5b84b/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_circleonxaxisarea.svg" style="margin: 10px 25px 25px 25px" width="350px"/>
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<p>
Set up the integral in polar coordinates for the area of the shaded region. Use the convention [mathjaxinline]\, 0\leq \theta \leq 2\pi[/mathjaxinline].<br/></p>
<p>
(Note that all three answer boxes are graded together, so they are either all correct or all incorrect.) </p>
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<col style="width:10%"/>
<col style="width:90%"/>
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<td colspan="2">
<div id="inputtype_polar-tab12-problem1_2_1" class=" capa_inputtype textline">
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<td>
<p> \( \displaystyle \huge{ \int }\)</p>
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<td style="padding-top: 28px">
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<span class="trailing_text" id="trailing_text_polar-tab12-problem1_2_2"> [mathjaxinline] d\theta [/mathjaxinline]</span>
<span class="status unanswered" id="status_polar-tab12-problem1_2_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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\(\)
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<p>
Evaluate the integral. <br/>(Enter your answer in terms of [mathjaxinline]\, \alpha ,\,[/mathjaxinline] and [mathjaxinline]\, a\,[/mathjaxinline] in the box below.)<br/></p>
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\(\)
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<p>
Verify your answer by computing the area directly without using an integral.<br/></p>
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<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
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<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
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<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
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<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
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<h2 class="hd hd-2 unit-title">13. More area examples</h2>
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Equivalent integrals
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Which of the following is equal to the area of a half circle with radius [mathjaxinline]a[/mathjaxinline]? (Check all that apply.)<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_polar-tab13-problem1_2_1">
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<input type="checkbox" name="input_polar-tab13-problem1_2_1[]" id="input_polar-tab13-problem1_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="polar-tab13-problem1_2_1-choice_1-label" for="input_polar-tab13-problem1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_polar-tab13-problem1_2_1"> <text>[mathjaxinline]\displaystyle \frac12\int _{\pi /2}^{\pi } 4a^2\cos ^2(\theta )\, d\theta[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab13-problem1_2_1[]" id="input_polar-tab13-problem1_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="polar-tab13-problem1_2_1-choice_2-label" for="input_polar-tab13-problem1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_polar-tab13-problem1_2_1"> <text>[mathjaxinline]\displaystyle \frac12\int _{\pi }^{\pi /2} 4a^2\cos ^2(\theta )\, d\theta[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab13-problem1_2_1[]" id="input_polar-tab13-problem1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="polar-tab13-problem1_2_1-choice_3-label" for="input_polar-tab13-problem1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_polar-tab13-problem1_2_1"> <text>[mathjaxinline]\displaystyle \frac12\int _{-\pi /2}^{0} 4a^2\cos ^2(\theta )\, d\theta[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab13-problem1_2_1[]" id="input_polar-tab13-problem1_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="polar-tab13-problem1_2_1-choice_4-label" for="input_polar-tab13-problem1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_polar-tab13-problem1_2_1"> <text>[mathjaxinline]\displaystyle \frac12\int _{-\pi /4}^{\pi /4} 4a^2\cos ^2(\theta )\, d\theta[/mathjaxinline]</text>
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Area inside a spiral
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<div class="caption"><b>Figure 20</b>: <span> Graph of [mathjaxinline]r=\theta \,[/mathjaxinline] for [mathjaxinline]\, 0\leq \theta \leq 4\pi[/mathjaxinline]<br/></span></div>
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Find the area of the shaded region.<br/>(Be careful not to overcount.)<br/></p>
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<p style="display:inline">Area [mathjaxinline]\, =\,[/mathjaxinline]</p>
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<font color="#0078b0">2520</font>
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<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
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<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
</tr>
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<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
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<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
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<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
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<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
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<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
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<h2 class="hd hd-2 unit-title">14. Graphing polar functions</h2>
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Another rose
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<p>
Sketch the polar curve given by [mathjaxinline]\, r=2\sin (4\theta )\,[/mathjaxinline] for [mathjaxinline]\, 0\leq \theta &lt; 2\pi[/mathjaxinline].<br/></p>
<p>
In your graph: </p>
<ul class="itemize">
<li>
<p>
Plot all the points [mathjaxinline]\, (r,\theta )\,[/mathjaxinline] at which [mathjaxinline]\, r=2\sin (4\theta )\,[/mathjaxinline] is maximum, using the "Max [mathjaxinline]r[/mathjaxinline] Points" tool </p>
</li>
<li>
<p>
Plot all the points [mathjaxinline]\, (r,\theta )\,[/mathjaxinline] at which [mathjaxinline]\, r=2\sin (4\theta )\,[/mathjaxinline] is minimum, using the "Min [mathjaxinline]r[/mathjaxinline] Points" tool </p>
</li>
<li>
<p>
Draw the rays at which [mathjaxinline]\, r=0\,[/mathjaxinline], using the "Guiding Rays" tool </p>
</li>
<li>
<p>
Sketch the curve [mathjaxinline]\, r=2\sin (4\theta )\,[/mathjaxinline] for [mathjaxinline]\, 0\leq \theta &lt; 2\pi[/mathjaxinline], using the "Polar Curve" tool </p>
</li>
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<p>
You will get the score if your points at which [mathjaxinline]r[/mathjaxinline] is maximum or minimum are at the right locations, regardless of whether your polar curve is correct. Drawing the curve is good practice for you, and will help us develop appropriate tolerances to grade these types of problems in the future. </p>
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Area of one leaf
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<p>
Here is the graph of [mathjaxinline]\, r=\sin (m\theta )\,[/mathjaxinline] where [mathjaxinline]\, m\geq 0[/mathjaxinline] is an even integer.<br/></p>
<div class="figure">
<center>
<img alt="See text above" src="/assets/courseware/v1/9d55f1477c3b05839baa405d8326c1ad/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_32leaf.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
<br/>
</center>
</div>
<p>
Determine [mathjaxinline]\, m\,[/mathjaxinline] and find the area of one leaf of the graph, shaded orange above.<br/></p>
<p>
<p style="display:inline">[mathjaxinline]m\, =\,[/mathjaxinline]</p>
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\(\)
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<p>
<p style="display:inline">Area of one leaf [mathjaxinline]\, =\,[/mathjaxinline]</p>
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Odd number of leaves
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<p>
Sketch the graph of [mathjaxinline]\, r=3\sin (3\theta )\,[/mathjaxinline] for [mathjaxinline]\, 0\, \leq \theta &lt; \pi[/mathjaxinline].<br/></p>
<p>
In your graph: </p>
<ul class="itemize">
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<p>
Plot all the points [mathjaxinline]\, (r,\theta )\,[/mathjaxinline] at which [mathjaxinline]\, r=3\sin (3\theta )\,[/mathjaxinline] is maximum, using the "Max [mathjaxinline]r[/mathjaxinline] Points" tool </p>
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<p>
Plot all the points [mathjaxinline]\, (r,\theta )\,[/mathjaxinline] at which [mathjaxinline]\, r=3\sin (3\theta )\,[/mathjaxinline] is minimum, using the "Min [mathjaxinline]r[/mathjaxinline] Points" tool </p>
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<p>
Draw the rays at which [mathjaxinline]\, r=0\,[/mathjaxinline], using the "Guiding Rays" tool </p>
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Sketch the curve [mathjaxinline]\, r=3\sin (3\theta )\,[/mathjaxinline] for [mathjaxinline]\, 0\leq \theta &lt; \pi[/mathjaxinline], using the "Polar Curve" tool </p>
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You get a point for attempting the problem, regardless of whether your submission is correct or not. It is good practice for you, and will help us develop appropriate tolerances to grade these types of problems in the future. <br/></p>
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You will receive feedback for the points at which [mathjaxinline]r[/mathjaxinline] is maximum or minimum, but not for the guiding rays or the polar curve.<br/></p>
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Number of leaves
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Consider the polar equation [mathjaxinline]\, r=\sin (3\theta ),\,[/mathjaxinline] which is similar to the equation in the previous problem, but now double the domain so that [mathjaxinline]\, 0\, \leq \theta \leq 2\pi[/mathjaxinline].<br/></p>
<p>
In the box below, enter the number of different leaves in the graph of [mathjaxinline]\, r=\sin (3\theta ),\,[/mathjaxinline] for [mathjaxinline]\, 0\, \leq \theta \leq 2\pi[/mathjaxinline].<br/><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_polar-tab14-problem4_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<p>
Now, for any <b class="bf">odd</b> positive integer [mathjaxinline]\, n,\,[/mathjaxinline] find the number of different leaves in the graph of [mathjaxinline]\, r=\sin (n\theta ),\,[/mathjaxinline] for [mathjaxinline]\, 0\, \leq \theta \leq 2\pi[/mathjaxinline].<br/>(Enter your answer in terms of [mathjaxinline]\, n\,[/mathjaxinline] below)<br/><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_polar-tab14-problem4_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<h2 class="hd hd-2 unit-title">15. More polar functions</h2>
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Choose a polar function from the drop down menu. Then use the sliders to adjust the values of [mathjaxinline]A[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and [mathjaxinline]n[/mathjaxinline]. You can restrict the plot to any subinterval of [mathjaxinline][0, 10\pi ][/mathjaxinline] using the blue slider. The orange point depicts [mathjaxinline](r(\theta ),\theta )[/mathjaxinline]; the value of [mathjaxinline]\theta[/mathjaxinline] can be adjusted using the orange slider. </p><iframe src="/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/PolarGraphing3.html" width="820 px" height="650 px" style="border:0px" scrolling="no" seamless="seamless"/>
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Period of rose: integer
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<p>
Consider the graph of [mathjaxinline]\, r(\theta )\, =\, \sin \left(n\theta \right)\,[/mathjaxinline] for [mathjaxinline]\, n&gt;0\,[/mathjaxinline] an integer.<br/></p>
<p>
You can investigate the graph using the Mathlet above with [mathjaxinline]\, r=A+B\sin \left(n\theta \right),\,[/mathjaxinline] [mathjaxinline]A=0,\,[/mathjaxinline] and [mathjaxinline]\, N\,[/mathjaxinline] an integer. <br/></p>
<p>
Notice that if you use the blue slider to increase the domain [mathjaxinline]\, 0\leq \theta \leq \, \theta _{\text {max}}\,[/mathjaxinline] of the graph from [mathjaxinline]\, \theta _{\text {max}}=0,\,[/mathjaxinline] there is a value of [mathjaxinline]\, \theta _{\text {max}}\,[/mathjaxinline] at which the graph is completed and does not change anymore even if we further increase the domain. We want to find when this happens. <br/><br/></p>
<p>
Find the smallest positive integer [mathjaxinline]\, k\,[/mathjaxinline] such that the graph of [mathjaxinline]\, r=\sin \left(n\theta \right)\,[/mathjaxinline] repeats itself every time [mathjaxinline]\, \theta \,[/mathjaxinline] increases by [mathjaxinline]\, k\pi[/mathjaxinline].<br/></p>
<p>
In other words, find the smallest integer [mathjaxinline]\, k&gt;0\,[/mathjaxinline] such that for all [mathjaxinline]\, \theta ,\,[/mathjaxinline] the point represented by [mathjaxinline]\, \left(r(\theta +k\pi ), \theta +k\pi \right)\,[/mathjaxinline] is the same as the point represented by [mathjaxinline]\, \left(r(\theta ), \theta \right)[/mathjaxinline].<br/></p>
<p><i class="itshape">Hint</i>: Do not forget that </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000462" style="table-layout:auto" width="100%">
<tr id="a0000000463">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \left(r(\theta ), \, \theta \right)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(-r(\theta ), \, \theta +\pi \right).[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.273)</td>
</tr>
</table>
<p>
You will find that the smallest [mathjaxinline]\, k\,[/mathjaxinline] may depend on whether [mathjaxinline]\, n\,[/mathjaxinline] is odd or even.<br/></p>
<p>
<p style="display:inline">For [mathjaxinline]\, n\,[/mathjaxinline] <b class="bf">odd</b>, [mathjaxinline]\, k=\,[/mathjaxinline] </p>
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<p style="display:inline">For [mathjaxinline]\, n\,[/mathjaxinline] <b class="bf">even</b>, [mathjaxinline]\, k=\,[/mathjaxinline]</p>
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Period of rose: odd denominator
</h3>
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<p>
Consider the graph of [mathjaxinline]\displaystyle \, r=\sin \left(\frac{n}{5}\theta \right)\,[/mathjaxinline] for any integer [mathjaxinline]\, n\,[/mathjaxinline] not a multiple of [mathjaxinline]\, 5[/mathjaxinline].<br/></p>
<p>
You can investigate the graph using the Mathlet above with [mathjaxinline]\, r=A+B\sin \left(n\theta \right),\,[/mathjaxinline] [mathjaxinline]A=0,\,[/mathjaxinline] and [mathjaxinline]\, N=0.2,\, \, 0.4,\, \, 0.6,\ldots[/mathjaxinline].<br/></p>
<p>
As above, you will notice that if you use the blue slider to increase the domain [mathjaxinline]\, 0\leq \theta \leq \, \theta _{\text {max}}\,[/mathjaxinline] of the graph from [mathjaxinline]\, \theta _{\text {max}}=0,\,[/mathjaxinline] there is a value of [mathjaxinline]\, \theta _{\text {max}}\,[/mathjaxinline] at which the graph is completed and does not change anymore even if we further increase the domain. We want to find when this happens. <br/><br/></p>
<p>
Find the smallest positive integer [mathjaxinline]\, k\,[/mathjaxinline] such that the graph of [mathjaxinline]\displaystyle \, r=\sin \left(\frac{n}{5}\theta \right)\,[/mathjaxinline] repeats itself every time [mathjaxinline]\, \theta \,[/mathjaxinline] increases by [mathjaxinline]\, k\pi[/mathjaxinline].<br/></p>
<p>
In other words, as in the previous problem, find the smallest integer [mathjaxinline]\, k&gt;0\,[/mathjaxinline] such that for all [mathjaxinline]\, \theta ,\,[/mathjaxinline] the point represented by [mathjaxinline]\, \left(r(\theta +k\pi ), \theta +k\pi \right)\,[/mathjaxinline] is the same as the point represented by [mathjaxinline]\, \left(r(\theta ), \theta \right)[/mathjaxinline].<br/></p>
<p><i class="itshape">Hint</i>: As above, do not forget that </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000475" style="table-layout:auto" width="100%">
<tr id="a0000000476">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \left(r(\theta ), \, \theta \right)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(-r(\theta ), \, \theta +\pi \right).[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.282)</td>
</tr>
</table>
<p>
You will find that the smallest [mathjaxinline]\, k\,[/mathjaxinline] may depend on whether [mathjaxinline]\, n\,[/mathjaxinline] is odd or even.<br/></p>
<p>
<p style="display:inline">For [mathjaxinline]\, n\,[/mathjaxinline] <b class="bf">odd</b>, [mathjaxinline]\, k=\,[/mathjaxinline] </p>
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<p style="display:inline">For [mathjaxinline]\, n\,[/mathjaxinline] <b class="bf">even</b>, [mathjaxinline]\, k=\,[/mathjaxinline]</p>
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Period of rose: Irrational
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<p>
As above, consider the graph of [mathjaxinline]\, r=\sin \left(a\theta \right).[/mathjaxinline]<br/></p>
<p>
However, this time, let [mathjaxinline]\, a\,[/mathjaxinline] be an irrational number, that is, [mathjaxinline]\, ak\,[/mathjaxinline] is <b class="bf">not</b> an integer for any integer [mathjaxinline]\, k\,[/mathjaxinline]. <br/></p>
<p>
Find the smallest positive integer [mathjaxinline]\, k\,[/mathjaxinline] such that the graph of [mathjaxinline]\, r=\sin \left(a\theta \right)\,[/mathjaxinline] repeats itself every time [mathjaxinline]\, \theta \,[/mathjaxinline] increases by [mathjaxinline]\, k\pi[/mathjaxinline].<br/></p>
<p>
In other words, as in the previous problem, find the smallest integer [mathjaxinline]\, k&gt;0\,[/mathjaxinline] such that for all [mathjaxinline]\, \theta ,\,[/mathjaxinline] the point represented by [mathjaxinline]\, \left(r(\theta +k\pi ), \theta +k\pi \right)\,[/mathjaxinline] is the same as the point represented by [mathjaxinline]\, \left(r(\theta ), \theta \right)[/mathjaxinline].<br/></p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_polar-tab15-problem3_2_1">
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<text> [mathjaxinline]\displaystyle k=\frac{1}{a}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle k=\frac{2}{a}[/mathjaxinline]</text>
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<text> No such [mathjaxinline]\, k\,[/mathjaxinline] exists: the graph never repeats itself as [mathjaxinline]\, \theta \,[/mathjaxinline] increases.</text>
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Period of the linear spiral
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Consider the graph of [mathjaxinline]\, r=n\theta ,[/mathjaxinline] for [mathjaxinline]\, n\neq 0[/mathjaxinline]. You can investigate this graph using the mathlet above.<br/></p>
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Find the smallest positive integer [mathjaxinline]\, k\,[/mathjaxinline] such that the graph of [mathjaxinline]\, r=n\theta ,[/mathjaxinline] repeats itself every time [mathjaxinline]\, \theta \,[/mathjaxinline] increases by [mathjaxinline]\, k\pi[/mathjaxinline].<br/></p>
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In other words, as in the previous problems, find the smallest integer [mathjaxinline]\, k&gt;0\,[/mathjaxinline] such that for all [mathjaxinline]\, \theta ,\,[/mathjaxinline] the point represented by [mathjaxinline]\, \left(r(\theta +k\pi ), \theta +k\pi \right)\,[/mathjaxinline] is the same as the point represented by [mathjaxinline]\, \left(r(\theta ), \theta \right)[/mathjaxinline].<br/></p>
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<text> [mathjaxinline]\displaystyle k=\frac{2\pi }{n}[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle k= \frac{1}{n}[/mathjaxinline]</text>
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<text> No such [mathjaxinline]\, k\,[/mathjaxinline] exists: the graph never repeats itself.</text>
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Recognize impossible equations
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<div class="caption"><b>Figure 25</b>: <span> The green region is the region for which [mathjaxinline]0.5\leq |r|\leq 10[/mathjaxinline]. Note the figure is not drawn to scale.<br/></span></div>
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Which of the following functions have graphs that lie completely in the green region? (Check all that apply.)<br/><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_polar-tab15-problem5_2_1">
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<input type="checkbox" name="input_polar-tab15-problem5_2_1[]" id="input_polar-tab15-problem5_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="polar-tab15-problem5_2_1-choice_1-label" for="input_polar-tab15-problem5_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_polar-tab15-problem5_2_1"> <text>[mathjaxinline]\displaystyle r=\sin \left(\frac{3\theta }{2}-\frac{\pi }{4}\right)[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab15-problem5_2_1[]" id="input_polar-tab15-problem5_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="polar-tab15-problem5_2_1-choice_2-label" for="input_polar-tab15-problem5_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_polar-tab15-problem5_2_1"> <text>[mathjaxinline]r=-1+2\cos (4\theta )[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab15-problem5_2_1[]" id="input_polar-tab15-problem5_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="polar-tab15-problem5_2_1-choice_3-label" for="input_polar-tab15-problem5_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_polar-tab15-problem5_2_1"> <text>[mathjaxinline]\displaystyle r=\frac{4}{\cos (\theta )}[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab15-problem5_2_1[]" id="input_polar-tab15-problem5_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="polar-tab15-problem5_2_1-choice_4-label" for="input_polar-tab15-problem5_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_polar-tab15-problem5_2_1"> <text>[mathjaxinline]r=-4+2.5\cos (\sqrt {2}\theta )[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">16. Graphing polar functions</h2>
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Practice Graphing
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<p>
Sketch the graph of [mathjaxinline]\, r=-\sin (\theta )+1\,[/mathjaxinline] for [mathjaxinline]\, 0\leq \theta &lt;2\pi[/mathjaxinline].<br/></p>
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Plot all the points [mathjaxinline]\, (r,\theta )\,[/mathjaxinline] at which [mathjaxinline]\, r=-\sin (\theta )+1\,[/mathjaxinline] is maximum, using the "Max [mathjaxinline]r[/mathjaxinline] Points" tool </p>
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<p>
Plot all the points [mathjaxinline]\, (r,\theta )\,[/mathjaxinline] at which [mathjaxinline]\, r=-\sin (\theta )+1\,[/mathjaxinline] is minimum, using the "Min [mathjaxinline]r[/mathjaxinline] Points" tool </p>
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<li>
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Sketch the curve [mathjaxinline]\, r=-\sin (\theta )+1\,[/mathjaxinline] for [mathjaxinline]\, 0\leq \theta &lt; 2\pi[/mathjaxinline], using the "Polar Curve" tool </p>
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(You can plot any number of point you like using the "Guiding Points" tool to help you sketch the curve.)<br/></p>
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You get a point for attempting the problem, regardless of whether your submission is correct or not. It is good practice for you, and will help us develop appropriate tolerances to grade these types of problems in the future. </p>
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Area of Hyperbola
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Set up the integral for the area [mathjaxinline]\, A\,[/mathjaxinline] in the first quadrant bounded by the [mathjaxinline]x[/mathjaxinline]-axis, the [mathjaxinline]\, y[/mathjaxinline]-axis, and the polar curve given by </p>
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[mathjaxinline]\displaystyle \displaystyle r[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{1+2\cos (\theta )} \quad \text {where}\, \frac{-2\pi }{3}&lt;\theta &lt; \frac{2\pi }{3}.[/mathjaxinline]
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\(\)
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<p style="display:inline; text-align:right"> \( \displaystyle \huge{ \int }\)</p>
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<span class="trailing_text" id="trailing_text_polar-tab17-problem1_3_1"> [mathjaxinline] d\theta[/mathjaxinline]</span>
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<h3 class="hd hd-3 problem-header" id="polar-tab17-problem2-problem-title" aria-describedby="block-v1:MITx+18.01.3x+1T2020+type@problem+block@polar-tab17-problem2-problem-progress" tabindex="-1">
Conic sections
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<p>
Consider the polar equation [mathjaxinline]\, \displaystyle r(\theta )=\frac{1}{1+b\cos (\theta )}[/mathjaxinline]. </p>
<p>
Find the values of [mathjaxinline]b[/mathjaxinline] for which the graph of [mathjaxinline]r=\, r(\theta )\,[/mathjaxinline] is<br/></p>
<table cellspacing="0" class="tabular" style="table-layout:auto">
<tr>
<td style="text-align:left; border:none">
an ellipse (or circle):</td>
<td style="text-align:left; border:none">
a parabola:</td>
<td style="text-align:left; border:none">
a hyperbola:</td>
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(Check all that apply.)</td>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_polar-tab17-problem2_2_1">
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<text>[mathjaxinline]b&lt;-1[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_2_1[]" id="input_polar-tab17-problem2_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="polar-tab17-problem2_2_1-choice_1-label" for="input_polar-tab17-problem2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_2_1">
<text>[mathjaxinline]b=-1[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_2_1[]" id="input_polar-tab17-problem2_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="polar-tab17-problem2_2_1-choice_2-label" for="input_polar-tab17-problem2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_2_1">
<text>[mathjaxinline]-1&lt;b&lt;0[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_polar-tab17-problem2_2_1[]" id="input_polar-tab17-problem2_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="polar-tab17-problem2_2_1-choice_3-label" for="input_polar-tab17-problem2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_2_1">
<text>[mathjaxinline]b=0[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_polar-tab17-problem2_2_1[]" id="input_polar-tab17-problem2_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="polar-tab17-problem2_2_1-choice_4-label" for="input_polar-tab17-problem2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_2_1">
<text>[mathjaxinline]0&lt; b &lt;1[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_2_1[]" id="input_polar-tab17-problem2_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="polar-tab17-problem2_2_1-choice_5-label" for="input_polar-tab17-problem2_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_2_1">
<text>[mathjaxinline]b=1[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_2_1[]" id="input_polar-tab17-problem2_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="polar-tab17-problem2_2_1-choice_6-label" for="input_polar-tab17-problem2_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_2_1">
<text>[mathjaxinline]b&gt;1[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_3_1[]" id="input_polar-tab17-problem2_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="polar-tab17-problem2_3_1-choice_0-label" for="input_polar-tab17-problem2_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_3_1">
<text>[mathjaxinline]b&lt;-1[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_3_1[]" id="input_polar-tab17-problem2_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="polar-tab17-problem2_3_1-choice_1-label" for="input_polar-tab17-problem2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_3_1">
<text>[mathjaxinline]b=-1[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_3_1[]" id="input_polar-tab17-problem2_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="polar-tab17-problem2_3_1-choice_2-label" for="input_polar-tab17-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_3_1">
<text>[mathjaxinline]-1&lt;b&lt;0[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_3_1[]" id="input_polar-tab17-problem2_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="polar-tab17-problem2_3_1-choice_3-label" for="input_polar-tab17-problem2_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_3_1">
<text>[mathjaxinline]b=0[/mathjaxinline]</text>
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<text>[mathjaxinline]0&lt; b &lt;1[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_3_1[]" id="input_polar-tab17-problem2_3_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="polar-tab17-problem2_3_1-choice_5-label" for="input_polar-tab17-problem2_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_3_1">
<text>[mathjaxinline]b=1[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_3_1[]" id="input_polar-tab17-problem2_3_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="polar-tab17-problem2_3_1-choice_6-label" for="input_polar-tab17-problem2_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_3_1">
<text>[mathjaxinline]b&gt;1[/mathjaxinline]</text>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<text>[mathjaxinline]b&lt;-1[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_4_1[]" id="input_polar-tab17-problem2_4_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="polar-tab17-problem2_4_1-choice_1-label" for="input_polar-tab17-problem2_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_4_1">
<text>[mathjaxinline]b=-1[/mathjaxinline]</text>
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<input type="checkbox" name="input_polar-tab17-problem2_4_1[]" id="input_polar-tab17-problem2_4_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="polar-tab17-problem2_4_1-choice_2-label" for="input_polar-tab17-problem2_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_4_1">
<text>[mathjaxinline]-1&lt;b&lt;0[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_polar-tab17-problem2_4_1[]" id="input_polar-tab17-problem2_4_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="polar-tab17-problem2_4_1-choice_3-label" for="input_polar-tab17-problem2_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_4_1">
<text>[mathjaxinline]b=0[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_polar-tab17-problem2_4_1[]" id="input_polar-tab17-problem2_4_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="polar-tab17-problem2_4_1-choice_4-label" for="input_polar-tab17-problem2_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_4_1">
<text>[mathjaxinline]0\leq b &lt;1[/mathjaxinline]</text>
</label>
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<input type="checkbox" name="input_polar-tab17-problem2_4_1[]" id="input_polar-tab17-problem2_4_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="polar-tab17-problem2_4_1-choice_5-label" for="input_polar-tab17-problem2_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_4_1">
<text>[mathjaxinline]b=1[/mathjaxinline]</text>
</label>
</div>
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<input type="checkbox" name="input_polar-tab17-problem2_4_1[]" id="input_polar-tab17-problem2_4_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="polar-tab17-problem2_4_1-choice_6-label" for="input_polar-tab17-problem2_4_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_polar-tab17-problem2_4_1">
<text>[mathjaxinline]b&gt;1[/mathjaxinline]</text>
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<h3 class="hd hd-3 problem-header" id="polar-tab17-problem3-problem-title" aria-describedby="block-v1:MITx+18.01.3x+1T2020+type@problem+block@polar-tab17-problem3-problem-progress" tabindex="-1">
Graph of ellipse
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<p>
In this problem, we investigate briefly the ellipse given by </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000520" style="table-layout:auto" width="100%">
<tr id="a0000000521">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle r[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac{1}{1+0.5\cos (\theta )},\qquad 0\leq \theta &lt;2\pi .[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.310)</td>
</tr>
</table>
<p>
Find the Cartesian coordinates of the center of the ellipse.<br/>(Enter your answer as an ordered pair surrounded by parentheses, and separated by a comma: e.g. "(a, b)". Enter coordinates as fractions or decimals to 2 decimal places.)<br/><p style="display:inline">The center of the ellipse is at [mathjaxinline]\, (x,y)\, =\,[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_polar-tab17-problem3_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_polar-tab17-problem3_2_1" id="input_polar-tab17-problem3_2_1" aria-describedby="status_polar-tab17-problem3_2_1" value="" class="math"/>
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<div id="display_polar-tab17-problem3_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_polar-tab17-problem3_2_1_dynamath" name="input_polar-tab17-problem3_2_1_dynamath"/>
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</div></div> </p>
<p>
Find the Cartesian coordinates of the point at which [mathjaxinline]\, r\,[/mathjaxinline] is minimum.<br/>(Enter your answer as an ordered pair: e.g. "(a, b)". Enter coordinates as fractions or decimals to 2 decimal places.)<br/><p style="display:inline">[mathjaxinline]r\,[/mathjaxinline] is minimum at [mathjaxinline]\, (x,y)\, =\,[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_polar-tab17-problem3_3_1" class="text-input-dynamath capa_inputtype inline textline">
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Find the Cartesian coordinates of the point at which [mathjaxinline]\, r\,[/mathjaxinline] is maximum.<br/>(Enter your answer as an ordered pair: e.g. "(a, b)". Enter coordinates as fractions or decimals to 2 decimal places.)<br/><p style="display:inline">[mathjaxinline]r\,[/mathjaxinline] is maximum at [mathjaxinline]\, (x,y)\, =\,[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_polar-tab17-problem3_4_1" class="text-input-dynamath capa_inputtype inline textline">
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<h2 class="hd hd-2 unit-title">18. Summary</h2>
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<p><b class="bfseries">Definition</b></p><div class="figure"><center><img src="/assets/courseware/v1/1180f1f8c684e720a7ba9920a139a0e5/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_intro2.svg" width="350px" alt="" style="margin: 10px 25px 25px 25px"/><br/></center></div><p>
The <span style="color:#27408C"><b class="bf">polar coordinates</b></span> of a point [mathjaxinline]\, P\,[/mathjaxinline] are an ordered pair [mathjaxinline]\, (r,\theta )\,[/mathjaxinline] such that </p><table id="a0000000531" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000532"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =r\cos (\theta )[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.317)</td></tr><tr id="a0000000533"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =r\sin (\theta ),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.318)</td></tr></table><p>
where the ordered pair [mathjaxinline]\, (x, y)\,[/mathjaxinline] give the rectangular coordinates of the point [mathjaxinline]P[/mathjaxinline].<br/></p><p>
In other words, given the polar coordinates [mathjaxinline]\, (r,\theta )\,[/mathjaxinline] of a point, we can find its [mathjaxinline]\, x[/mathjaxinline]- and [mathjaxinline]\, y[/mathjaxinline]-coordinates using these formulas. </p><p>
The usual rectangular coordinates are also called <span style="color:#27408C"><b class="bf">Cartesian coordinates</b></span>. To find polar coordinates from Cartesian coordinates, we use </p><table id="a0000000534" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000535"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle r[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\pm \sqrt {x^2+y^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.319)</td></tr><tr id="a0000000536"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \theta[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, \arctan \left(\frac{y}{x}\right).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.320)</td></tr></table><p>
However, [mathjaxinline]\, r\,[/mathjaxinline] and [mathjaxinline]\, \theta \,[/mathjaxinline] are not unique for any given point, as explained below.<br/></p><p>
Polar coordinates are motivated by the fact that we can locate a point on a plane by specifying: </p><table id="a0000000537" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000538"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle r:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \text {the distance from the origin to the point,}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.321)</td></tr><tr id="a0000000539"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \theta :[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \text {the angle of the ray from the origin to the point with the positive}\, x\text {-axis}.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.322)</td></tr></table><p><b class="bfseries">Ambiguities in polar coordinates</b></p><p>
The polar coordinates describing a point are not unique. First, </p><table id="a0000000540" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000541"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle (r,\theta )[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(r, \theta + 2\pi n\right) \qquad (n\, \text {any integer}).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.323)</td></tr></table><p>
That is , knowing the [mathjaxinline]x[/mathjaxinline]- and [mathjaxinline]y[/mathjaxinline]- coordinates only determines [mathjaxinline]\theta[/mathjaxinline] up to [mathjaxinline]2\pi[/mathjaxinline]-periodicity. We frequently use conventions such as: </p><table id="a0000000542" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000543"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 0\leq \theta < 2\pi[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.324)</td></tr><tr id="a0000000544"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {or}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle -\pi < \theta \leq \pi .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.325)</td></tr></table><p>
Second, </p><table id="a0000000545" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000546"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle (-r,\theta )[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(r, \theta \pm \pi \right) \qquad -\infty <r<\infty[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.326)</td></tr><tr id="a0000000547"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {and equivalently}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle (r,\theta )[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \left(-r, \theta \pm \pi \right) \qquad -\infty <r<\infty .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.327)</td></tr></table><p><b class="bfseries">Finding theta</b></p><p>
To find [mathjaxinline]\, \theta ,[/mathjaxinline] we first find </p><table id="a0000000548" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000549"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \theta _0[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\arctan \left(\frac{|y|}{|x|}\right).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.328)</td></tr></table><div id="a0000000550" class="figure"><center><img src="/assets/courseware/v1/2c8c43872a73b75e26d26e284a322dbf/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_theta0.svg" width="350px" alt="See caption" style="margin: 10px 25px 25px 25px"/><div class="caption"><b>Figure 28</b>: <span> Here, [mathjaxinline](x,y)\,[/mathjaxinline] lies in the second quadrant, [mathjaxinline]\, \theta =\pi -\theta _0\,[/mathjaxinline]. </span></div></center></div><p>
Then we find [mathjaxinline]\, \theta \,[/mathjaxinline] using [mathjaxinline]\, \theta _0,\,[/mathjaxinline] by considering which quadrant it lies in, which is best done using a picture like the one above. </p><p><b class="bfseries">Circles and rays in polar coordinates</b></p><p>
An equation in polar coordinates is called a <span style="color:#27408C"><b class="bf">polar equation</b></span>. We will mostly be dealing with polar equations of the form [mathjaxinline]\, r=r(\theta )[/mathjaxinline]. </p><p>
The simplest examples are: </p><ul class="itemize"><li><p>
[mathjaxinline]r=a[/mathjaxinline]: </p><div class="figure"><center><img src="/assets/courseware/v1/60360aea8787841c7da914903200a10c/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_circle.svg" width="350px" alt="" style="margin: 10px 25px 25px 25px"/><br/></center></div></li><li><p>
[mathjaxinline]\theta = \alpha[/mathjaxinline] ([mathjaxinline]r\geq 0[/mathjaxinline]):<br/></p><div class="figure"><center><img src="/assets/courseware/v1/63b0357df06f8bb36a4b3f39ea3c9abe/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_ray.svg" width="350px" alt="" style="margin: 10px 25px 25px 25px"/><br/></center></div><p>
If we use the convention [mathjaxinline]-\infty <r<\infty ,\,[/mathjaxinline] then [mathjaxinline]\, \theta =\alpha \,[/mathjaxinline] is a line through the origin: </p><div class="figure"><center><img src="/assets/courseware/v1/3c75c1c037502bbe88e74bb3cc488854/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_line.svg" width="350px" alt="" style="margin: 10px 25px 25px 25px"/><br/></center></div></li></ul><p>
We use the circles and rays as the grid of the polar coordinate systems. </p><p><b class="bfseries">Rotation about the origin</b></p><p>
The graph of [mathjaxinline]\, r=r(\theta -\alpha )\,[/mathjaxinline] is obtained by rotating the graph of [mathjaxinline]\, r=r(\theta )\,[/mathjaxinline] <b class="bf">about the origin</b> by the angle [mathjaxinline]\, +\alpha[/mathjaxinline].<br/></p><p>
If [mathjaxinline]\, \alpha >0\,[/mathjaxinline] the rotation is counterclockwise.<br/></p><p>
if [mathjaxinline]\, \alpha <0\,[/mathjaxinline] the rotation is clockwise.<br/></p><p><b class="bfseries">The area element</b></p><div id="a0000000551" class="figure"><center><img src="/assets/courseware/v1/5ea0c62a82f73be2e77fc338fec4793f/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_dA.svg" width="350px" alt="See caption" style="margin: 10px 25px 25px 25px"/><br/><div class="caption"><b>Figure 29</b>: <span><table id="a0000000552" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000553"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \Delta A[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \approx \frac{1}{2} r^2 \Delta \theta[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.329)</td></tr></table></span></div></center></div><p>
Passing to the differential, the area element [mathjaxinline]\, dA\,[/mathjaxinline] in polar coordinates is the area of the infinitesimal region spanned by [mathjaxinline]d\theta[/mathjaxinline], given by </p><table id="a0000000554" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000555"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle d A[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{2} r(\theta )^2 d\theta .[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.330)</td></tr></table><div class="figure"><center><img src="/assets/courseware/v1/f4aa4b23083cc1c75faa743177fbc0cb/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_polar_areaIntegral.svg" width="400px" alt="" style="margin: 10px 25px 25px 25px"/><br/></center></div><p>
The area bounded a curve [mathjaxinline]r=r(\theta )[/mathjaxinline], and the two rays [mathjaxinline]\, \theta =\theta _1\,[/mathjaxinline] and [mathjaxinline]\, \theta =\theta _2\,[/mathjaxinline] is </p><table id="a0000000556" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000557"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle A[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =\int _{\theta _1}^{\theta _2} d A[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \frac{1}{2}\int _{\theta _1}^{\theta _2} r(\theta )^2 d\theta[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.331)</td></tr></table><p>
Examples of polar curves </p><table id="a0000000558" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000559"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \text {Lines}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.332)</td></tr><tr id="a0000000560"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y=b:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r\,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, \frac{b}{\sin (\theta )} \qquad (0<\theta <\pi )[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.333)</td></tr><tr id="a0000000561"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x=b:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r\,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, \frac{b}{\cos (\theta )}\qquad (-\pi /2<\theta <\pi /2)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.334)</td></tr><tr id="a0000000562"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle y=mx+b :[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r\,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, \frac{b}{\sin (\theta )-m\cos (\theta )}\qquad \left(\arctan (m) <\theta <\arctan (m)+\pi \right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.335)</td></tr><tr id="a0000000563"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Circles touching the origin}\, :\,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.336)</td></tr><tr id="a0000000564"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {centered on }\, x\, \text {-axis}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r\,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, 2b\cos (\theta )\qquad (-\pi /2\leq \theta \leq \pi /2)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.337)</td></tr><tr id="a0000000565"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {centered on }\, y\, \text {-axis}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r\,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, 2b\sin (\theta )\qquad (0\leq \theta <\pi )[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.338)</td></tr><tr id="a0000000566"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {centered on }\, \theta =\alpha \, \, \text {ray}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r\,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, 2b\cos (\theta -\alpha )\qquad (\alpha -\pi /2\leq \theta <\alpha +\pi /2)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.339)</td></tr><tr id="a0000000567"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Roses/Limacons/Cardioids (see Mathlet)}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r\,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, A+B\cos (n\theta )\qquad (\text {Domain varies})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.340)</td></tr><tr id="a0000000568"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r\,[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\, A+B\sin (n \theta )\qquad (\text {Domain varies})[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.341)</td></tr><tr id="a0000000569"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Spirals}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =n\theta \qquad (0\leq \theta )[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.342)</td></tr><tr id="a0000000570"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Conics with one focus at the origin}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.343)</td></tr><tr id="a0000000571"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Hyperbolas}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\frac{1}{1+b\cos (\theta )}, \, \, |b|>1\qquad (0\leq \theta <2\pi , 1+b\cos (\theta )\neq 0 )[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.344)</td></tr><tr id="a0000000572"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Parabolas}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\frac{1}{1+b\cos (\theta )}, \, \, |b|=1\qquad (0\leq \theta <2\pi , 1+b\cos (\theta )\neq 0 )[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.345)</td></tr><tr id="a0000000573"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Ellipses}:[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle r[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\frac{1}{1+b\cos (\theta )},\, \, |b|<1\, \qquad (0\leq \theta <2\pi )[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.346)</td></tr></table>
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For a printable version, go <a href="https://courses.edx.org/asset-v1:MITx+18.01.3x+1T2020+type@asset+block@polar_summary.pdf" target="_blank">here</a>. </p>
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