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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Surface area</h3>
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<h2 class="hd hd-2 unit-title">2. Arc length and surface area</h2>
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<p><h3>Objectives</h3></p><ul class="itemize"><li><p>
Set up (and evaluate) integrals to find the <span style="color:#27408C"><b class="bf">arc length</b></span> of curves. </p></li><li><p>
Set up (and evaluate) integrals to find the <span style="color:#27408C"><b class="bf">surface area</b></span> of surfaces of rotation. </p></li></ul><p><h3>Contents: 8 pages</h3></p><p>
7 videos (64 minutes at 1x speed) 16 problems </p>
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<h2 class="hd hd-2 unit-title">3. Arc length</h2>
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What is the length of the line [mathjaxinline]y=1-x/2[/mathjaxinline] for [mathjaxinline]0 \leq x \leq 2[/mathjaxinline]? </p>
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<p>
Consider a very small piece of a curve [mathjaxinline]\Delta s[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/29642c71a3c91d4f8d0110089999fc1b/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_arclength_im2.svg" width="300px" alt="A right triangle with legs of length Delta y and Delta x. The the length of the hypotenuse is given by the square root of Delta x squared plus Delta y squared. The hypotenuse is approximately the same length as Delta s." style="margin: 10px 25px 25px 25px"/></center><p>
From the image, we see that the length of the small piece of curve is nearly the same as the length of the secant line. </p><table id="a0000002648" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002649"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \Delta s[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {(\Delta x)^2 + (\Delta y)^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.581)</td></tr></table><p>
Thus as we pass to the differential, this approximate equality becomes an equality and we are left with </p><table id="a0000002650" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002651"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle ds[/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 \sqrt {(dx)^2 + (dy)^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.582)</td></tr></table><p>
The differential arc length element [mathjaxinline]ds[/mathjaxinline] can be written in terms of [mathjaxinline]dx[/mathjaxinline] and [mathjaxinline]dy[/mathjaxinline] in the following equivalent ways for a curve [mathjaxinline]y=f(x)[/mathjaxinline]: </p><table id="a0000002652" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002653"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle ds[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \sqrt {dx^2 + dy^2}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</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">(4.583)</td></tr><tr id="a0000002654"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \displaystyle \sqrt {1 + \left(\frac{dy}{dx}\right)^2}\, dx[/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 \displaystyle \sqrt {1 + \left(f'(x) \right)^2}\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.584)</td></tr><tr id="a0000002655"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \displaystyle \sqrt { \left(\frac{dx}{dy}\right)^2 + 1}\, dy[/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 \displaystyle \sqrt {\left((f^{-1})'(y) \right)^2 + 1}\, dy[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.585)</td></tr></table><center><img src="/assets/courseware/v1/e56baf571bbbdf53ecef5a9a240d0151/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_arclength_im2b.svg" width="300px" alt="A curve is plotted in the first quadrant. The left endpoint has coordinates x sub 0 comma y sub 0 and is labeled s sub 0. The right endpoint has coordinates x sub n comma y sub n and is labeled s sub n." style="margin: 10px 25px 25px 25px"/></center><p>
The <span style="color:#27408C"><b class="bf">arc length</b></span>, s, of the curve [mathjaxinline]y=f(x)[/mathjaxinline] over the interval shown is the integral: </p><table id="a0000002656" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002657"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Arc length } s[/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 \int _{s_0}^{s_ n} ds[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.586)</td></tr><tr id="a0000002658"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \int _{x_0}^{x_ n} \sqrt {1+f'(x)^2}\, dx, \qquad y=f(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.587)</td></tr><tr id="a0000002659"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \int _{y_0}^{y_ n} \sqrt {1+\left((f^{-1})'(y)\right)^2}\, dy, \qquad x=f^{-1}(y).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.588)</td></tr></table>
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Practice finding ds 1
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Given the curve [mathjaxinline]y=\sin x[/mathjaxinline] for [mathjaxinline]0 \leq x \leq \pi /2[/mathjaxinline], find the arc length element [mathjaxinline]ds[/mathjaxinline] as a differential in terms of [mathjaxinline]dx[/mathjaxinline]. </p>
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(Enter answer in terms of [mathjaxinline]x[/mathjaxinline] only.) </p>
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<p style="display:inline">[mathjaxinline]ds=[/mathjaxinline]</p>
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Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
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Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
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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">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">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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Practice finding ds 2
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Given the curve [mathjaxinline]y=\sin x[/mathjaxinline] for [mathjaxinline]0 \leq x \leq \pi /2[/mathjaxinline], find the arc length element [mathjaxinline]ds[/mathjaxinline] as a differential in terms of [mathjaxinline]dy[/mathjaxinline]. </p>
<p>
(Enter answer in terms of [mathjaxinline]y[/mathjaxinline] only.) </p>
<p>
<p style="display:inline">[mathjaxinline]ds=[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_technique5-tab3-problem3_2_1"> [mathjaxinline]dy[/mathjaxinline]</span>
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<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>
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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>
</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>
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<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>
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<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>
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<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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Practice finding ds 3
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Given the curve [mathjaxinline]x^2+y^3= x[/mathjaxinline], find the arc length element [mathjaxinline]ds[/mathjaxinline] as a differential in terms of [mathjaxinline]dx[/mathjaxinline]. </p>
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(Enter answer in terms of [mathjaxinline]x[/mathjaxinline] only.) </p>
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<p style="display:inline">[mathjaxinline]ds=[/mathjaxinline]</p>
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<span class="trailing_text" id="trailing_text_technique5-tab3-problem4_2_1"> [mathjaxinline]dx[/mathjaxinline]</span>
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<h2 class="hd hd-2 unit-title">4. Setting up arc length integrals</h2>
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Setting up an arc length integral 1
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Set up the integral to find the arc length of the curve [mathjaxinline]y=e^ x \sin x[/mathjaxinline] for [mathjaxinline]0 \leq x \leq \pi /2[/mathjaxinline]. </p>
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(Do not solve. Enter the integrand with respect to [mathjaxinline]x[/mathjaxinline]. )</p>
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<p> \( \displaystyle \Large{ \int_0^{\pi/2} }\)</p>
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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 <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
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</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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Setting up an arc length integral 2
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Set up the integral to find the arc length of the curve [mathjaxinline]y=\tan x[/mathjaxinline] for [mathjaxinline]0 \leq x \leq \pi /4[/mathjaxinline]. </p>
<p>
(Do not solve. Enter the limits and integrand with respect to [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]dx[/mathjaxinline]. Note that all three answer boxes are graded together: they are either graded all correct or all wrong.)<br/></p>
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<p> \( \displaystyle \huge{ \int }\)</p>
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<span class="trailing_text" id="trailing_text_technique5-tab4-problem2_2_2"> [mathjaxinline] dx [/mathjaxinline]</span>
<span class="status unanswered" id="status_technique5-tab4-problem2_2_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<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>
<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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Setting up an arc length integral 3
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Set up the integral to find the arc length of the curve [mathjaxinline]x=\ln (\sec y)[/mathjaxinline] for [mathjaxinline]0 \leq y \leq \pi /4[/mathjaxinline]. </p>
<p>
(Do not solve. Enter the limits and integrand with respect to [mathjaxinline]y[/mathjaxinline] and [mathjaxinline]dy[/mathjaxinline] only. Note that all three answer boxes are graded together: they are either graded all correct or all wrong.)<br/></p>
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<p> \( \displaystyle \huge{ \int }\)</p>
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<input type="text" name="input_technique5-tab4-problem3_2_2" id="input_technique5-tab4-problem3_2_2" data-input-id="technique5-tab4-problem3_2_2" value="" aria-describedby="trailing_text_technique5-tab4-problem3_2_2 status_technique5-tab4-problem3_2_2" size="10"/>
<span class="trailing_text" id="trailing_text_technique5-tab4-problem3_2_2"> [mathjaxinline] dy [/mathjaxinline]</span>
<span class="status unanswered" id="status_technique5-tab4-problem3_2_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<span class="status unanswered" id="status_technique5-tab4-problem3_2_3" data-tooltip="Not yet answered.">
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Evaluate the integral that computes the arc length of the parabola: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \int _0^ a \sqrt {1+4x^2}\, dx,[/mathjax]</td>
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using the trig substitution [mathjaxinline]\displaystyle x=\frac12 \tan y[/mathjaxinline], [mathjaxinline]\displaystyle dx = \frac12 \sec ^2 y\, dy[/mathjaxinline]. </p>
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Compute the arc length
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[mathjaxinline]\displaystyle y = (1/3)\left(2+x^2\right)^{3/2}[/mathjaxinline], &#8195;[mathjaxinline]1\leq x \leq 2[/mathjaxinline]. </p>
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Consider the surface area created by rotating the curve [mathjaxinline]y=f(x)[/mathjaxinline] about the [mathjaxinline]x[/mathjaxinline]-axis. </p><center><img src="/assets/courseware/v1/f66863dcd6da57f9dfae242d92640230/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_arclength_im3.svg" width="300px" alt="A curve is plotted in the first quadrant. The left endpoint has coordinates x sub 0 comma y sub 0 and is labeled s sub 0. The right endpoint has coordinates x sub n comma y sub n and is labeled s sub n. A segment of the curve is labeled Delta s. This segment is rotated about the x-axis, forming a vertical ring." style="margin: 10px 25px 25px 25px"/></center><p>
If you imagine cutting the band, the area is approximately equal to the area of the rectangle whose base is the circumference [mathjaxinline]2\pi y[/mathjaxinline], and whose height is [mathjaxinline]\Delta s[/mathjaxinline]. The differential area element is [mathjaxinline]\displaystyle dA = (2\pi y) ds.[/mathjaxinline] </p><p>
The surface area is computed by the integral: </p><table id="a0000002725" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]A = \displaystyle \int _{s_0}^{s_ n} (2\pi y)\, ds.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><center><img src="/assets/courseware/v1/fed3a659efaa0596e108b139c732206d/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_arclength_im4.svg" width="500px" alt="A curve is plotted in the first quadrant. The left endpoint has coordinates x sub 0 comma y sub 0 and is labeled s sub 0. The right endpoint has coordinates x sub n comma y sub n and is labeled s sub n. A segment of the curve is labeled Delta s. This segment is rotated about the y-axis, forming a horizontal ring." style="margin: 10px 25px 25px 25px"/></center><p>
Note that if we take a curve and rotate about the [mathjaxinline]y[/mathjaxinline]-axis, the differential area element is [mathjaxinline]\displaystyle dA = (2\pi x) ds,[/mathjaxinline] and the surface area is computed by the integral </p><table id="a0000002726" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]A = \displaystyle \int _{s_0}^{s_ n} (2\pi x)\, ds.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Note that because the differential [mathjaxinline]ds[/mathjaxinline] can be described in terms of [mathjaxinline]dx[/mathjaxinline] or [mathjaxinline]dy[/mathjaxinline], we can set up either of these surface area integrals with respect to [mathjaxinline]dx[/mathjaxinline] or [mathjaxinline]dy[/mathjaxinline]. </p><p><span style="color:#99182C"><b class="bf">Warning:</b></span> Make sure that your limits of integration make sense depending on whether you are integrating with respect to [mathjaxinline]x[/mathjaxinline] or [mathjaxinline]y[/mathjaxinline]. </p>
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Consider the surface obtained by rotating the curve [mathjaxinline]y=1/x[/mathjaxinline] for [mathjaxinline]1 \leq x \leq 3[/mathjaxinline] about the <b class="bf">[mathjaxinline]x[/mathjaxinline]-axis</b>. Set up the integrals for the surface area of this surface in two ways, integrating with respect to [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. </p>
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</td>
</tr>
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</span>
<p>
(Do not solve. Enter the integrand with respect to [mathjaxinline]y[/mathjaxinline] only.) </p>
<span>
<table>
<col style="width:10%"/>
<col style="width:90%"/>
<tbody>
<tr>
<td>
<p> \( \displaystyle \Large{ \int_{1/3}^1 }\)</p>
</td>
<td style="padding-top: 28px">
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<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_technique5-tab6-problem1_3_1" class="inputtype formulaequationinput">
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<span class="trailing_text" id="trailing_text_technique5-tab6-problem1_3_1"> [mathjaxinline] \large{dy}. [/mathjaxinline]</span>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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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>
<tr class="formulainput">
<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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Setting up surface area integrals 2
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<p>
Consider the surface obtained by rotating the curve [mathjaxinline]y=e^{-x}[/mathjaxinline] for [mathjaxinline]0 \leq x \leq 2[/mathjaxinline] about the <b class="bf">[mathjaxinline]x[/mathjaxinline]-axis</b>. In this problem you will practice setting up the integral to integrate both with respect to [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]y[/mathjaxinline]. </p>
<p>
(Do not solve. Enter the limits and integrand with respect to [mathjaxinline]x[/mathjaxinline] only. Note that all three answer boxes are graded together.) </p>
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<td>
<p> \( \displaystyle \huge{ \int }\)</p>
</td>
<td>
<br/>
<div id="formulaequationinput_technique5-tab6-problem2_2_2" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique5-tab6-problem2_2_2" id="input_technique5-tab6-problem2_2_2" data-input-id="technique5-tab6-problem2_2_2" value="" aria-describedby="trailing_text_technique5-tab6-problem2_2_2 status_technique5-tab6-problem2_2_2" size="40"/>
<span class="trailing_text" id="trailing_text_technique5-tab6-problem2_2_2"> [mathjaxinline] dx [/mathjaxinline]</span>
<span class="status unanswered" id="status_technique5-tab6-problem2_2_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_technique5-tab6-problem2_2_2" class="answer"/>
<div id="input_technique5-tab6-problem2_2_2_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
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<p>
(Do not solve. This time, find the limits and integrand with respect to [mathjaxinline]y[/mathjaxinline] only. All three answer boxes are graded together.)<br/></p>
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<td>
<p> \( \displaystyle \huge{ \int }\)</p>
</td>
<td>
<br/>
<div id="formulaequationinput_technique5-tab6-problem2_3_2" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
<div class="unanswered">
<input type="text" name="input_technique5-tab6-problem2_3_2" id="input_technique5-tab6-problem2_3_2" data-input-id="technique5-tab6-problem2_3_2" value="" aria-describedby="trailing_text_technique5-tab6-problem2_3_2 status_technique5-tab6-problem2_3_2" size="30"/>
<span class="trailing_text" id="trailing_text_technique5-tab6-problem2_3_2"> [mathjaxinline] dy [/mathjaxinline]</span>
<span class="status unanswered" id="status_technique5-tab6-problem2_3_2" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div id="input_technique5-tab6-problem2_3_2_preview" class="equation">
\(\)
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<div class="formulainput">
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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>
<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>
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<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
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Setting up surface area integrals 3
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Consider the surface obtained by rotating the curve [mathjaxinline]y=1-x^2[/mathjaxinline] for [mathjaxinline]0 \leq x \leq 1[/mathjaxinline] about the <b class="bf">[mathjaxinline]y[/mathjaxinline]-axis</b>. Set up the integral for the surface area of the surface as an integral with respect to [mathjaxinline]y[/mathjaxinline]. </p>
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(Do not solve. Enter the limits and integrand with respect to [mathjaxinline]y[/mathjaxinline] only. Note that all three answer boxes are graded together.)<br/></p>
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<span class="trailing_text" id="trailing_text_technique5-tab6-problem3_2_2"> [mathjaxinline] dy [/mathjaxinline]</span>
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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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<h2 class="hd hd-2 unit-title">7. Surface area of a sphere</h2>
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Compute the surface area 1
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Consider the surface obtained by rotating the segment of the curve [mathjaxinline]y = 1-2x[/mathjaxinline] in the first quadrant about the <b class="bf">[mathjaxinline]x[/mathjaxinline]-axis</b>. <br/>(The first quadrant of the [mathjaxinline]x,y-[/mathjaxinline] plane consists of all points with [mathjaxinline]\, x,y \geq 0[/mathjaxinline].)<br/></p>
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<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">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
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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">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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<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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<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>
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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">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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Compute the surface area 2
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Consider the surface obtained by rotating the curve [mathjaxinline]y = x^2[/mathjaxinline], [mathjaxinline]0\le x \le \sqrt {2}[/mathjaxinline] around the <b class="bf">[mathjaxinline]y[/mathjaxinline]-axis</b>. </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"><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>
</tr>
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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>
<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-2">Worked example: surface area of a torus</h3>
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Consider the surface obtained by rotating the curve [mathjaxinline]y=1-x^2[/mathjaxinline] for [mathjaxinline]0 \leq x \leq 1[/mathjaxinline] about the <b class="bf">[mathjaxinline]y[/mathjaxinline]-axis</b>. (Note you set up this integral on the previous page.) </p>
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<h2 class="hd hd-2 unit-title">8. Summary</h2>
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<p><b class="bfseries">Arc length</b></p><p>
Consider a very small piece of a curve [mathjaxinline]\Delta s[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/29642c71a3c91d4f8d0110089999fc1b/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_arclength_im2.svg" width="300px" alt="A right triangle with legs of length Delta y and Delta x. The the length of the hypotenuse is given by the square root of Delta x squared plus Delta y squared. The hypotenuse is approximately the same length as Delta s." style="margin: 10px 25px 25px 25px"/></center><p>
From the image, we see that the length of the small piece of curve is nearly the same as the length of the secant line. </p><table id="a0000002772" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002773"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \Delta s[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \approx[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sqrt {(\Delta x)^2 + (\Delta y)^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.660)</td></tr></table><p>
Thus as we pass to the differential, this approximate equality becomes an equality and we are left with </p><table id="a0000002774" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002775"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle ds[/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 \sqrt {(dx)^2 + (dy)^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.661)</td></tr></table><p>
The differential arc length element [mathjaxinline]ds[/mathjaxinline] can be written in terms of [mathjaxinline]dx[/mathjaxinline] and [mathjaxinline]dy[/mathjaxinline] in the following equivalent ways for a curve [mathjaxinline]y=f(x)[/mathjaxinline]: </p><table id="a0000002776" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002777"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle ds[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \sqrt {dx^2 + dy^2}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</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">(4.662)</td></tr><tr id="a0000002778"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \displaystyle \sqrt {1 + \left(\frac{dy}{dx}\right)^2}\, dx[/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 \displaystyle \sqrt {1 + \left(f'(x) \right)^2}\, dx[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.663)</td></tr><tr id="a0000002779"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \displaystyle \sqrt { \left(\frac{dx}{dy}\right)^2 + 1}\, dy[/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 \displaystyle \sqrt {\left((f^{-1})'(y) \right)^2 + 1}\, dy[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.664)</td></tr></table><center><img src="/assets/courseware/v1/e56baf571bbbdf53ecef5a9a240d0151/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_arclength_im2b.svg" width="300px" alt="A curve is plotted in the first quadrant. The left endpoint has coordinates x sub 0 comma y sub 0 and is labeled s sub 0. The right endpoint has coordinates x sub n comma y sub n and is labeled s sub n." style="margin: 10px 25px 25px 25px"/></center><p>
The <span style="color:#27408C"><b class="bf">arc length</b></span>, s, of the curve [mathjaxinline]y=f(x)[/mathjaxinline] over the interval shown is the integral: </p><table id="a0000002780" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000002781"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Arc length } s[/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 \int _{s_0}^{s_ n} ds[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.665)</td></tr><tr id="a0000002782"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \int _{x_0}^{x_ n} \sqrt {1+f'(x)^2}\, dx, \qquad y=f(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.666)</td></tr><tr id="a0000002783"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \int _{y_0}^{y_ n} \sqrt {1+\left((f^{-1})'(y)\right)^2}\, dy, \qquad x=f^{-1}(y).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(4.667)</td></tr></table><p><b class="bfseries">Surface area</b></p><p>
Consider the surface area created by rotating the curve [mathjaxinline]y=f(x)[/mathjaxinline] about the [mathjaxinline]x[/mathjaxinline]-axis. </p><center><img src="/assets/courseware/v1/f66863dcd6da57f9dfae242d92640230/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_arclength_im3.svg" width="300px" alt="A curve is plotted in the first quadrant. The left endpoint has coordinates x sub 0 comma y sub 0 and is labeled s sub 0. The right endpoint has coordinates x sub n comma y sub n and is labeled s sub n. A segment of the curve is labeled Delta s. This segment is rotated about the x-axis, forming a vertical ring." style="margin: 10px 25px 25px 25px"/></center><p>
If you imagine cutting the band, the area is approximately equal to the area of the rectangle whose base is the circumference [mathjaxinline]2\pi y[/mathjaxinline], and whose height is [mathjaxinline]\Delta s[/mathjaxinline]. The differential area element is [mathjaxinline]\displaystyle dA = (2\pi y) ds.[/mathjaxinline] </p><p>
The surface area is computed by the integral: </p><table id="a0000002784" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]A = \displaystyle \int _{s_0}^{s_ n} (2\pi y)\, ds.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><center><img src="/assets/courseware/v1/fed3a659efaa0596e108b139c732206d/asset-v1:MITx+18.01.2x+3T2019+type@asset+block/images_arclength_im4.svg" width="500px" alt="A curve is plotted in the first quadrant. The left endpoint has coordinates x sub 0 comma y sub 0 and is labeled s sub 0. The right endpoint has coordinates x sub n comma y sub n and is labeled s sub n. A segment of the curve is labeled Delta s. This segment is rotated about the y-axis, forming a horizontal ring." style="margin: 10px 25px 25px 25px"/></center><p>
Note that if we take a curve and rotate about the [mathjaxinline]y[/mathjaxinline]-axis, the differential area element is [mathjaxinline]\displaystyle dA = (2\pi x) ds,[/mathjaxinline] and the surface area is computed by the integral </p><table id="a0000002785" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]A = \displaystyle \int _{s_0}^{s_ n} (2\pi x)\, ds.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
Note that because the differential [mathjaxinline]ds[/mathjaxinline] can be described in terms of [mathjaxinline]dx[/mathjaxinline] or [mathjaxinline]dy[/mathjaxinline], we can set up either of these surface area integrals with respect to [mathjaxinline]dx[/mathjaxinline] or [mathjaxinline]dy[/mathjaxinline]. </p><p><span style="color:#99182C"><b class="bf">Warning:</b></span> Make sure that your limits of integration make sense depending on whether you are integrating with respect to [mathjaxinline]x[/mathjaxinline] or [mathjaxinline]y[/mathjaxinline]. </p>
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