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<h2 class="hd hd-2 unit-title">1. Motivation</h2>
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<h3 class="hd hd-2">Parametric equations</h3>
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<h2 class="hd hd-2 unit-title">2. Parametric equations</h2>
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<p><b class="bfseries">Objectives</b></p><ul class="itemize"><li><p>
Describe curves using <span style="color:#27408C"><b class="bf">parametric equations</b></span>. </p></li><li><p><span style="color:#27408C"><b class="bf">Sketch parametric curves</b></span> by plotting points or <span style="color:#27408C"><b class="bf">eliminating the parameter</b></span>, and determining the direction. </p></li><li><p>
Find the <span style="color:#27408C"><b class="bf">speed</b></span> of a particle whose <span style="color:#27408C"><b class="bf">position</b></span> is described by a parametric curve. </p></li><li><p>
Do <span style="color:#27408C"><b class="bf">calculus</b></span> with parametric curves: </p><ul class="itemize"><li><p>
Find the <span style="color:#99182C"><b class="bf">tangent line</b></span> of a parametric curve. </p></li><li><p>
Find the <span style="color:#99182C"><b class="bf">arc length</b></span> of a parametric curve. </p></li><li><p>
Find the <span style="color:#99182C"><b class="bf">surface area</b></span> generated by rotating a parametric curve. </p></li><li><p>
Find the <span style="color:#99182C"><b class="bf">area</b></span> of a region bounded by a parametric curve. </p></li></ul></li></ul><p><b class="bfseries">Contents: 15 pages</b></p><p>
10 videos (54 minutes 1x speed) 23 questions </p>
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<h2 class="hd hd-2 unit-title">3. Preparation</h2>
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The circle
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<p>
You are familiar with the circle of radius 1, and the implicit function that defines it [mathjaxinline]x^2+y^2=1[/mathjaxinline]. </p>
<p>
If you are on the top half of the circle, what is the expression for [mathjaxinline]y[/mathjaxinline] in terms of [mathjaxinline]x[/mathjaxinline]? </p>
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<p style="display:inline">[mathjaxinline]y=[/mathjaxinline]</p>
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\(\)
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If you are on the left half of the circle, what is the expression for [mathjaxinline]x[/mathjaxinline] in terms of [mathjaxinline]y[/mathjaxinline]? </p>
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<p style="display:inline">[mathjaxinline]x=[/mathjaxinline]</p>
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\(\)
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<th class="formulainput" scope="col">Example Entries</th>
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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>
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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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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">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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<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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<p><b class="bfseries">Parametric curves</b></p><p>
A parametric curve (in the plane) is a curve defined by two equations </p><table id="a0000000006" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000007"><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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x(t),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.1)</td></tr><tr id="a0000000008"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y(t),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.2)</td></tr></table><p>
where [mathjaxinline]t[/mathjaxinline] is called a <span style="color:#27408C"><b class="bf">parameter</b></span>. For each real number [mathjaxinline]t[/mathjaxinline], the point [mathjaxinline](x(t), y(t))[/mathjaxinline] is a point on the curve. </p>
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Clockwise circular motion
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Based on the example in the video, what is a parameterization for the unit circle centered at the origin that starts at the point [mathjaxinline](1,0)[/mathjaxinline] when [mathjaxinline]t=0[/mathjaxinline] and moves <span style="color:#99182C"><b class="bf">clockwise</b></span> around the circle, returning to [mathjaxinline](1,0)[/mathjaxinline] when [mathjaxinline]t=2\pi[/mathjaxinline]? </p>
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<p style="display:inline">[mathjaxinline]x(t)=[/mathjaxinline]</p>
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Eliminate parameters practice 1
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Find the implicit equation for the curve given by the parametric equations [mathjaxinline]x(t) = t + t^2[/mathjaxinline] and [mathjaxinline]y(t) = t + 2t^2[/mathjaxinline], for [mathjaxinline]t[/mathjaxinline] any real number. </p>
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<text> [mathjaxinline]y^2 - 2xy + 2x^2 - y + x = 0[/mathjaxinline]</text>
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<text> [mathjaxinline]y^2 - 4xy + 4x^2 - 4y + 4x = 0[/mathjaxinline]</text>
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Eliminate parameters practice 2
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Find the implicit equation for the curve given by the parametric equations </p>
<table cellpadding="7" cellspacing="0" class="equation" id="a0000000018" style="table-layout:auto" width="100%">
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<td class="equation" style="width:80%; border:none">[mathjax](x(t), y(t)) = (\tan (t), \sec (t)) \rlap {\ ,}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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for [mathjaxinline]t[/mathjaxinline] in the intervals [mathjaxinline](-\pi /2, \pi /2)[/mathjaxinline] and [mathjaxinline](\pi /2, 3\pi /2)[/mathjaxinline]? </p>
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<text> [mathjaxinline]x^2 - y^2 = 1[/mathjaxinline]</text>
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<text> [mathjaxinline]y^2 - x^2 = 2[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">5. Sketching parametric curves</h2>
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Finding points on a parametric curve
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Eliminating parameters isn't always easy or obvious. Let's see if we can get a sense of a parametric curve by plotting some points. </p>
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Suppose that a particle follows the path defined by the parametric curve defined by </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000022" style="table-layout:auto" width="100%">
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle x(t)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle 2\cos t - \cos 2t[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.7)</td>
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[mathjaxinline]\displaystyle y(t)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle 2\sin t - \sin 2t.[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.8)</td>
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Find the position of the particle for each time indicated. </p>
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(Enter your answer as an ordered pair surrounded by parentheses, and separated by a comma: e.g. (a, b). Enter as math expressions or as decimals with at least 2 decimal places of accuracy.) </p>
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<p style="display:inline">[mathjaxinline]t=0[/mathjaxinline]: &#8195;&#8201; &#8201;</p>
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Where is the particle at the time [mathjaxinline]t=\pi /4[/mathjaxinline]? </p>
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<p style="display:inline">[mathjaxinline]t=\pi /4[/mathjaxinline]: &#8195;</p>
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<p style="display:inline">[mathjaxinline]t=\pi /2[/mathjaxinline]: &#8195;</p>
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<p style="display:inline">[mathjaxinline]t=\pi[/mathjaxinline]: &#8195;&#8201; &#8201;</p>
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<p><b class="bfseries">Exercise</b></p><p>
Try plotting the following parametric curves. </p><ol class="enumerate"><li value="1"><p>
[mathjaxinline]x(t) = t^2[/mathjaxinline], [mathjaxinline]y(t) = 2t[/mathjaxinline] </p></li><li value="2"><p>
[mathjaxinline]x(t) = 3\cos t[/mathjaxinline], [mathjaxinline]y(t) = 2\sin t[/mathjaxinline] </p></li><li value="3"><p>
[mathjaxinline]x(t) = t\cos t[/mathjaxinline], [mathjaxinline]y(t) = t\sin t[/mathjaxinline] </p></li></ol><p>
Use the mathlet below to check your answers! To get a sense of how parametric equations describe trajectories in the plane, click the >> button to watch the curve draw in for selected values of [mathjaxinline]t[/mathjaxinline]. </p><iframe src="https://mathlets1801.surge.sh/parametricCurves.html" width="1100 px" height="630 px" style="border:0px; display: block;"/>
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In the video that follows, one of the arguments relies on the chain rule. Let's review what the chain rule says. </p>
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Consider the composite function [mathjaxinline]f(g(t))[/mathjaxinline]. Then according to the chain rule: </p>
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Tangent lines preparation
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Consider the segment of the parametric curve [mathjaxinline]x=x(t)[/mathjaxinline], &#8201; [mathjaxinline]y=y(t)[/mathjaxinline] drawn below. </p>
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The two marked points are [mathjaxinline]\left(x(t_0), y(t_0)\right)[/mathjaxinline] and [mathjaxinline]\left(x(t_1), y(t_1)\right)[/mathjaxinline] for two times [mathjaxinline]t_0 &lt; t_1[/mathjaxinline]. Let </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000031" style="table-layout:auto" width="100%">
<tr id="a0000000032">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta t[/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 t_1-t_0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.13)</td>
</tr>
<tr id="a0000000033">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta x[/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 x(t_1)-x(t_0)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.14)</td>
</tr>
<tr id="a0000000034">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Delta y[/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 y(t_1)-y(t_0)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.15)</td>
</tr>
</table>
<p>
Which expression best approximates the slope of the tangent line to the parametric curve at the point [mathjaxinline]\left(x(t_0), y(t_0)\right)[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">7. Tangent lines of parametric curves</h2>
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The slope of a parametric curve [mathjaxinline]x=x(t)[/mathjaxinline], [mathjaxinline]y=y(t)[/mathjaxinline] is </p><table id="a0000000040" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{dy}{dx} = \large {\frac{\left(\frac{dy}{dt}\right)}{\left(\frac{dx}{dt}\right)}}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In particular, to find the slope of the tangent line to the curve at [mathjaxinline]t=t_0[/mathjaxinline], we compute [mathjaxinline]\displaystyle \frac{y'(t_0)}{x'(t_0)}[/mathjaxinline]. </p>
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Where is the tangent line horizontal?
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Consider the parametric curve </p>
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \displaystyle x[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \cos t[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.19)</td>
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[mathjaxinline]\displaystyle \displaystyle y[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \frac12 \sin 2t.[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.20)</td>
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For which time(s) [mathjaxinline]0 \leq t &lt; 2\pi[/mathjaxinline] is the tangent line horizontal? </p>
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(If there is more than one time [mathjaxinline]t[/mathjaxinline], separate multiple answers with a comma; e.g. 1, 2, pi/2, 2*pi. Note do not forget to enter multiplication explicitly: [mathjaxinline]2\pi[/mathjaxinline] must be entered as 2*pi.) </p>
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Where is the tangent line vertical?
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As above, consider the parametric curve </p>
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<tr id="a0000000048">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle x[/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 \cos t[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.23)</td>
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<tr id="a0000000049">
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<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle y[/mathjaxinline]
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<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
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<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \frac12 \sin 2t.[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.24)</td>
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For which time(s) [mathjaxinline]0 \leq t &lt; 2\pi[/mathjaxinline] is the tangent line vertical? </p>
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(If there is more than one time [mathjaxinline]t[/mathjaxinline], separate multiple answers with a comma; e.g. 1, 2, pi/2.) </p>
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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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Find the slope of the tangent line at (0,0)
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As above, consider the parametric curve </p>
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \displaystyle x[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \cos t[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.27)</td>
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[mathjaxinline]\displaystyle \displaystyle y[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle \frac12 \sin 2t.[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.28)</td>
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Find the slope(s) of the tangent line(s) through the origin [mathjaxinline](x,y)=(0,0)[/mathjaxinline]. </p>
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(If there is more than one slope, separate multiple answers with a comma; e.g. 1, 2, pi/2.) </p>
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<td class="formulainput">+ - * / (add, subtract, multiply, divide)</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">_ (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>
</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 \( 2\pi \)
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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>
</tr>
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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>
<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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Sketch the curve
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Sketch the parametric curve </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000059" style="table-layout:auto" width="100%">
<tr id="a0000000060">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle x[/mathjaxinline]
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<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 \cos t[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.31)</td>
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \displaystyle y[/mathjaxinline]
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<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 \frac12 \sin 2t[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.32)</td>
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for [mathjaxinline]0 \leq t \leq 2\pi[/mathjaxinline] by plotting points and using the information about tangent lines that you found in the problems above. </p>
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You <b class="bf">get a point</b> 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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Use the Curve dropdown menu to choose between using the Freeform or Spline drawing tool. </p>
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You will receive feedback on the curve, the horizontal and vertical tangents, but not the guiding points. </p>
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<h2 class="hd hd-2 unit-title">8. Review arc length</h2>
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Consider a piece of curve of length [mathjaxinline]\Delta s[/mathjaxinline] pictured in blue in the figure below. </p>
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When the piece of curve is very small, the length of the secant line segment is a good approximation for [mathjaxinline]\Delta s[/mathjaxinline]. </p>
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What equation describes this approximation? </p>
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<text> [mathjaxinline]\Delta s \approx \left(\Delta x\right)^2 + \left(\Delta y\right)^2[/mathjaxinline]</text>
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<text> [mathjaxinline]\displaystyle \Delta s\approx \frac{1}{2} \Delta x \Delta y[/mathjaxinline]</text>
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What equation best describes what happens when we pass to the differential? </p>
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<text> [mathjaxinline]\displaystyle ds = \frac12 dx\, dy[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">9. Arc length of parametric curves</h2>
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<p><b class="bfseries">A note about notation</b></p><table id="a0000000064" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000065"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle ds^2 = dx^2 + dy^2[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \qquad \text {means} \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (ds)^2 = (dx)^2 + (dy)^2[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.34)</td></tr><tr id="a0000000066"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle ds = \sqrt {dx^2 + dy^2}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \qquad \text {means} \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle ds =\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">(1.35)</td></tr></table><p>
In particular, [mathjaxinline]\displaystyle dx^2 = (dx)^2.[/mathjaxinline] This is the square of a differential, not the differential of the square. The differential of the square is [mathjaxinline]\displaystyle d(x^2) = 2x\, dx[/mathjaxinline], which is not the same. </p>
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<h3 class="hd hd-2">Arc length of parametric curves</h3>
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<p>
Consider a particle moving along a trajectory. The motion is described by the parametric curve </p><table id="a0000000067" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000068"><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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.36)</td></tr><tr id="a0000000069"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y(t).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.37)</td></tr></table><p>
The speed of the particle is given by </p><table id="a0000000070" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000071"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{ds}{dt}[/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 {\left(x'(t)\right)^2+\left(y'(t)\right)^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.38)</td></tr></table><p>
The differential arc length element is given by </p><table id="a0000000072" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000073"><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 {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.39)</td></tr></table><p>
Consider the arc length, or distance travelled by the particle from time [mathjaxinline]t_0[/mathjaxinline] to time [mathjaxinline]t_1[/mathjaxinline]. The letter [mathjaxinline]s[/mathjaxinline] is customarily used to denote arc length. You should think of [mathjaxinline]s=s(t)[/mathjaxinline] as a function of time, where [mathjaxinline]s(t)[/mathjaxinline] is the distance travelled by the particle since some starting time. If [mathjaxinline]s_0 = s(t_0)[/mathjaxinline] and [mathjaxinline]s_1=s(t_1)[/mathjaxinline], then the distance traveled by the particle from time [mathjaxinline]t_0[/mathjaxinline] to [mathjaxinline]t_1[/mathjaxinline] can be calculated by </p><table id="a0000000074" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000075"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle s_1 - s_0 = \int _{s_0}^{s_1} ds = \int _{t_0}^{t_1} \sqrt {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.40)</td></tr></table>
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Express the length of the ellipse [mathjaxinline]x^2/a^2 + y^2/b^2 =1[/mathjaxinline] using the parametrization [mathjaxinline]x = a\cos t[/mathjaxinline] and [mathjaxinline]y = b\sin t[/mathjaxinline]. (Do not evaluate.) </p>
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<p><b class="bf">Video note:</b> In the video below at minute 5:27, the final line for [mathjaxinline]\, ds\,[/mathjaxinline] should read [mathjaxinline]\, \frac{\sqrt {2(t^4+1)}}{t^2} dt.\, \,[/mathjaxinline] Prof. Joel Lewis missed a factor of [mathjaxinline]\, 2\,[/mathjaxinline] inside the square root in the numerator, but the only effect in the following arc length computation is an overall missing factor of [mathjaxinline]\, \sqrt {2}[/mathjaxinline]. </p>
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Find the length of the curve [mathjaxinline]x= t^2[/mathjaxinline], [mathjaxinline]y = t^3[/mathjaxinline] for [mathjaxinline]0\le t \le 2[/mathjaxinline]. </p>
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<h2 class="hd hd-2 unit-title">10. Position and speed along a parametric curve</h2>
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<p>
We have been thinking of a parametric curve as the description of a particle's position over time. So what is the velocity? And what about the acceleration? </p><ul class="itemize"><li><p>
The derivative [mathjaxinline]x'(t)[/mathjaxinline] is the velocity in the direction of the [mathjaxinline]x[/mathjaxinline]-axis. </p></li><li><p>
The derivative [mathjaxinline]y'(t)[/mathjaxinline] is the velocity in the direction of the [mathjaxinline]y[/mathjaxinline]-axis. </p></li><li><p>
The speed along the curve is given by [mathjaxinline]\displaystyle \frac{ds}{dt} = \sqrt {\left(x'(t)\right)^2 + \left(y'(t)\right)^2}.[/mathjaxinline] </p></li><li><p>
The notion of velocity along the curve requires considering [mathjaxinline]x'[/mathjaxinline] and [mathjaxinline]y'[/mathjaxinline] together in what is known as a vector. Both velocity and acceleration are vectors and you will see them in multivariable calculus. </p></li></ul>
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Pumpkin toss problem
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Every year at Halloween, students toss pumpkins off of the Green Building (the tallest building on MIT campus). The Green building is 90 meters tall. The height [mathjaxinline]y(t)[/mathjaxinline] of the pumpkin satisfies </p>
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle y^{\prime \prime }(t)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle -9.8 \quad \text {meters/sec}^2[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.51)</td>
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[mathjaxinline]\displaystyle y'(0)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle 0.2 \quad \text {meters/sec}[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.52)</td>
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[mathjaxinline]\displaystyle y(0)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle 90 \quad \text {meters}[/mathjaxinline]
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The position of the pumpkin starts at the edge of the green building and moves with constant velocity (assume drag forces are negligible here). </p>
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[mathjaxinline]\displaystyle x'(t)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle 0.5 \quad \text {meters/sec}[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.54)</td>
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[mathjaxinline]\displaystyle x(0)[/mathjaxinline]
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[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle 0 \quad \text {meters}[/mathjaxinline]
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<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.55)</td>
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Find the parametric curve [mathjaxinline]\left(x(t),y(t) \right)[/mathjaxinline] for the trajectory of the pumpkin. </p>
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<p style="display:inline">[mathjaxinline]x(t)=[/mathjaxinline]</p>
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Find an expression in terms of [mathjaxinline]\, t\,[/mathjaxinline] for the speed of the pumpkin [mathjaxinline]\displaystyle \frac{ds}{dt}[/mathjaxinline] as it moves along the trajectory (and before it hits the ground). </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{ds}{dt}=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">11. Surface area</h2>
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<p><b class="bfseries">Review of 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><div id="a0000000107" class="figure"><center><img src="/assets/courseware/v1/a95843a63a1586dfeff1a87e292f1935/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_arclength_im3.svg" width="300px" alt="See caption" style="margin: 10px 25px 25px 25px"/><div class="caption"><b>Figure 1</b>: <span>Rotating a curve about the [mathjaxinline]x[/mathjaxinline]-axis.<br/></span></div></center></div><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 surface area element is [mathjaxinline]\displaystyle dA = (2\pi y) ds.[/mathjaxinline] </p><p>
The surface area is computed by the integral: </p><table id="a0000000108" 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><div id="a0000000109" class="figure"><center><img src="/assets/courseware/v1/cf7c568d5c7fb89a0f4f2fd708c4aa31/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_arclength_im4.svg" width="500px" alt="See caption" style="margin: 10px 25px 25px 25px"/><div class="caption"><b>Figure 2</b>: <span>Rotating a curve about the [mathjaxinline]y[/mathjaxinline]-axis.<br/></span></div></center></div><p>
Note that if we take a curve and rotate about the [mathjaxinline]y[/mathjaxinline]-axis, the differential surface area element is [mathjaxinline]\displaystyle dA = (2\pi x) ds,[/mathjaxinline] and the surface area is computed by the integral </p><table id="a0000000110" 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>
Now we will find the surface area for surfaces created by rotating a curve defined by a parametric equation. <br/></p><p><span style="color:#99182C"><b class="bf">Warning:</b></span> In the video that follows, there is a typo. Where it says [mathjaxinline]dA[/mathjaxinline], it should say [mathjaxinline]A[/mathjaxinline]. </p>
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<h3 class="hd hd-2">Surface area</h3>
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<p><b class="bfseries">Surface area</b></p><p>
Consider the parametric curve </p><table id="a0000000111" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000112"><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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.62)</td></tr><tr id="a0000000113"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y(t).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.63)</td></tr></table><p>
Consider the surface formed by rotating the curve about the [mathjaxinline]y[/mathjaxinline]-axis. </p><p>
The differential surface area element is given by </p><table id="a0000000114" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000115"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle dA[/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 2\pi x\, ds[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.64)</td></tr><tr id="a0000000116"><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 2\pi x(t) \sqrt {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.65)</td></tr></table><p>
The surface area from time [mathjaxinline]t_0[/mathjaxinline] to time [mathjaxinline]t_1[/mathjaxinline] is given by the integral: </p><table id="a0000000117" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000118"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Surface area of parametric curve} = \int _{t_0}^{t_1} 2\pi x(t) \sqrt {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.66)</td></tr></table><p>
Consider the surface formed by rotating the curve about the [mathjaxinline]x[/mathjaxinline]-axis. </p><p>
The differential surface area element is given by </p><table id="a0000000119" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000120"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle dA[/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 2\pi y\, ds[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.67)</td></tr><tr id="a0000000121"><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 2\pi y(t) \sqrt {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.68)</td></tr></table><p>
The surface area from time [mathjaxinline]t_0[/mathjaxinline] to time [mathjaxinline]t_1[/mathjaxinline] is given by the integral: </p><table id="a0000000122" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000123"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Surface area of parametric curve} = \int _{t_0}^{t_1} 2\pi y(t) \sqrt {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.69)</td></tr></table><p>
The following mathlet is the same as the one earlier in this section. It may be useful for the following problem. </p><iframe src="https://mathlets1801.surge.sh/parametricCurves.html" width="1100 px" height="630 px" style="border:0px; display: block;"/>
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Surface area practice
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Consider the curve defined by </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000124" style="table-layout:auto" width="100%">
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle x[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
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[mathjaxinline]\displaystyle 3\cos t + \cos 3t[/mathjaxinline]
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<tr id="a0000000126">
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[mathjaxinline]\displaystyle y[/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 3\sin t - \sin 3t[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.71)</td>
</tr>
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for [mathjaxinline]0 \leq t \leq 2\pi[/mathjaxinline]. </p>
<p>
Form a surface by rotating this curve about the [mathjaxinline]x[/mathjaxinline]-axis. (Note this curve is sketched in the mathlet.) </p>
<p>
Set up the integral to find the surface area. </p>
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<h2 class="hd hd-2 unit-title">12. Area under parametric curves</h2>
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The (signed) area under a curve [mathjaxinline]y=f(x)[/mathjaxinline] between [mathjaxinline]x=a[/mathjaxinline] and [mathjaxinline]x=b[/mathjaxinline] is given by [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx[/mathjaxinline]. </p><p>
Suppose that this curve is parameterized by the equations </p><table id="a0000000136" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000137"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle x[/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 x(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.78)</td></tr><tr id="a0000000138"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y(t).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.79)</td></tr></table><p>
Moreover suppose that [mathjaxinline]x(t_0) = a[/mathjaxinline] and [mathjaxinline]x(t_1) = b[/mathjaxinline]. Then the (signed) area under the curve is also equal to </p><table id="a0000000139" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000140"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _ a^ b f(x)\, 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 \int _{t_0}^{t_1} f(x(t))\, x'(t) \, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.80)</td></tr></table><p>
by change of variables (or by substitution and applying the chain rule). But note that </p><table id="a0000000141" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000142"><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 y[/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 f(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.81)</td></tr><tr id="a0000000143"><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 y[/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 y(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.82)</td></tr><tr id="a0000000144"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Longrightarrow[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle y(t)[/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 f(x(t))[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.83)</td></tr></table><p>
hence </p><table id="a0000000145" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000146"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _ a^ b f(x)\, 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 \int _{t_0}^{t_1} y(t)\, x'(t) \, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.84)</td></tr></table><p>
This formula for the (signed) area under a parametric curve holds in general! </p><p><b class="bfseries">General result for parametric curves</b></p><p>
Given a parametric curve: </p><table id="a0000000147" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000148"><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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.85)</td></tr><tr id="a0000000149"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y(t),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.86)</td></tr></table><p>
The signed area of the region bounded between the curve and the [mathjaxinline]x[/mathjaxinline]-axis for [mathjaxinline]t_0 < t < t_1[/mathjaxinline] is given by the integral </p><table id="a0000000150" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000151"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \textrm{Signed Area}[/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 _{t_0}^{t_1} y(t)\, x'(t) \, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.87)</td></tr></table>
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Consider the portion of the parabola parameterized by [mathjaxinline]x = \sqrt {t}[/mathjaxinline] and [mathjaxinline]y=t[/mathjaxinline] for [mathjaxinline]t&gt;0[/mathjaxinline]. </p>
<p>
Find the area under the parabola for [mathjaxinline]0 \leq x \leq 2[/mathjaxinline] </p>
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<p><b class="bfseries">Areas enclosed by parametric curves</b></p><p>
Note that we can extend this idea of finding area under curves to finding the area enclosed by parametric curves in the case that the graph forms a closed region. </p><p>
However, you must be very careful when thinking about signs, and in identifying the limits of integration for such regions! Take care. </p>
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Find the (unsigned) area bounded by the curve </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000157" style="table-layout:auto" width="100%">
<tr id="a0000000158">
<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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 3\cos t + \cos 3t[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.92)</td>
</tr>
<tr id="a0000000159">
<td style="width:40%; border:none">&#160;</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 =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle 3\sin t - \sin 3t[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.93)</td>
</tr>
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<p>
as [mathjaxinline]t[/mathjaxinline] ranges from [mathjaxinline]0[/mathjaxinline] to [mathjaxinline]2\pi[/mathjaxinline]. (Note this curve is sketched in the mathlet on the previous page.) </p>
<p>
(Hint: use the double angle and sum angle formulas. ) </p>
<p>
(Enter answer as a mathematical expression or as a decimal to 2 decimal places. Your answer should be a positive number.) </p>
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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>
</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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<h2 class="hd hd-2 unit-title">13. Finding parameterizations</h2>
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Finding parametric equations 1
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Consider the parabola given by [mathjaxinline]f(x) = x^2[/mathjaxinline]. Parameterize the parabola as functions [mathjaxinline](x(t),y(t))[/mathjaxinline], where [mathjaxinline]x(t)[/mathjaxinline] and [mathjaxinline]y(t)[/mathjaxinline] are functions of the parameter [mathjaxinline]t = f'(x)[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]x(t) =[/mathjaxinline]</p>
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<br/>
<p style="display:inline">[mathjaxinline]y(t) =[/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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<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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<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 \( (2\pi+y)dy \)
</td>
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Finding parametric equations 2
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Consider the implicit function [mathjaxinline]x^3+y^3 = 3xy[/mathjaxinline]. Set [mathjaxinline]y=xt[/mathjaxinline] to find parametric equations [mathjaxinline]x(t)[/mathjaxinline] and [mathjaxinline]y(t)[/mathjaxinline] that describe this curve. </p>
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<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>
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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>
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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 \(\lambda \)
</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>
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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>
<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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<p><b class="bfseries">Wheel mathlet</b></p><p>
The cycloid is depicted in the following mathlet. The case considered in the video is when [mathjaxinline]a=b[/mathjaxinline]. You can play with these parameters to see different types of cycloid curves. </p><iframe src="https://mathlets1801.surge.sh/wheel.html" width="1100 px" height="630 px" style="border:0px; display: block;"/>
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Cycloid
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<p>
Taking [mathjaxinline]\theta = t[/mathjaxinline] and [mathjaxinline]a=1[/mathjaxinline] we get the cycloid given parametrically by [mathjaxinline]x = t - \sin t[/mathjaxinline], [mathjaxinline]y = 1-\cos t[/mathjaxinline]. This parametric curve describes the path of a point [mathjaxinline]P[/mathjaxinline] on a rolling wheel. If [mathjaxinline]t[/mathjaxinline] represents time, then the wheel is rotating at a constant (unit) speed. </p>
<p>
(a)&#8195;What is the speed of the point [mathjaxinline]P[/mathjaxinline] at each time [mathjaxinline]t[/mathjaxinline]? </p>
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(b)&#8195;Find the largest and smallest horizontal velocity ([mathjaxinline]dx/dt[/mathjaxinline]) of the point [mathjaxinline]P[/mathjaxinline]. </p>
<p>
<p style="display:inline">Largest:</p>
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<p style="display:inline">Smallest:</p>
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(c)&#8195;Find the length of the cycloid for one turn of the wheel. (Hint: use a half angle formula.) </p>
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(d)&#8195;Set up the integral that computes the area under the cycloid for one turn of the wheel. </p>
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<p> \( \displaystyle \huge{ \int_0^{2\pi} }\)</p>
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<h2 class="hd hd-2 unit-title">14. Parametric curves and their graphs</h2>
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Comparing sketches of two parametric curves
</h3>
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<p>
Sketch the two parametric curves for [mathjaxinline]0 \leq t \leq 1/2[/mathjaxinline] in the region provided below. </p>
<p>
<b class="bf">Curve 1:</b>
</p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000197" style="table-layout:auto" width="100%">
<tr id="a0000000198">
<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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \cos (2\pi t)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.120)</td>
</tr>
<tr id="a0000000199">
<td style="width:40%; border:none">&#160;</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 =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sin (2\pi t)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.121)</td>
</tr>
</table>
<p>
<b class="bf">Curve 2:</b>
</p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000200" style="table-layout:auto" width="100%">
<tr id="a0000000201">
<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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \cos (2\pi t^2)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.122)</td>
</tr>
<tr id="a0000000202">
<td style="width:40%; border:none">&#160;</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 =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sin (2\pi t^2)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.123)</td>
</tr>
</table>
<p>
(Sketch Curve 1 in blue using the Curve 1 tool. Sketch Curve 2 in orange using the Curve 2 tool. You will receive feedback on the curve, but not the guiding points.) </p>
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<li value="1">
<p>
Use the Curve dropdown menu to choose between using the Freeform or Spline drawing tool. </p>
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<li>
<p>
The Freeform tool draws like a regular pencil using your mouse. </p>
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<li>
<p>
The Spline tool allows you to pick a series of discrete points, and it then automatically connects them (once you have two or more points) with a smooth curve. </p>
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<li value="2">
<p>
You can use the Guiding Point tool to plot points to help with the curve. </p>
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<p>
Use the Horizontal and Vertical tangent tools to indicate where the tangent lines are horizontal and vertical respectively. </p>
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You will receive feedback on the curve, the horizontal and vertical tangents, but not the guiding points. </p>
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Comparing speeds of two parametric curves
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Find the speeds of the two curves. <b class="bf">Curve 1:</b> </p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000203" style="table-layout:auto" width="100%">
<tr id="a0000000204">
<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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \cos (2\pi t)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.124)</td>
</tr>
<tr id="a0000000205">
<td style="width:40%; border:none">&#160;</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 =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sin (2\pi t)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.125)</td>
</tr>
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<p>
<b class="bf">Curve 2:</b>
</p>
<table cellpadding="7" cellspacing="0" class="eqnarray" id="a0000000206" style="table-layout:auto" width="100%">
<tr id="a0000000207">
<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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \cos (2\pi t^2)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.126)</td>
</tr>
<tr id="a0000000208">
<td style="width:40%; border:none">&#160;</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 =[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \sin (2\pi t^2)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td class="eqnnum" style="width:20%; border:none;text-align:right">(1.127)</td>
</tr>
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<p>
<p style="display:inline">Speed of curve 1: [mathjaxinline]\displaystyle \frac{ds}{dt}=[/mathjaxinline]</p>
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<p style="display:inline">Speed of curve 2: [mathjaxinline]\displaystyle \frac{ds}{dt}=[/mathjaxinline]</p>
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<p><b class="bfseries">Parametric curves and their graphs</b></p><p>
Observe that the same curve can be parameterized in different ways. </p><p>
Note that the graph of a parametric curve gives you <b class="bf">less information</b> than what is incorporated by the parametric equations themselves. Because the parameter [mathjaxinline]t[/mathjaxinline] is hidden, you don't see where you are at each time, or how fast the curve is traced out. In the previous problems, you saw this by comparing the speeds of two curves whose trajectories followed the same circle. </p>
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<h2 class="hd hd-2 unit-title">15. Summary</h2>
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<p><b class="bfseries">Parametric curves</b></p><p>
A parametric curve (in the plane) is a curve defined by two equations </p><table id="a0000000217" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000218"><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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x(t),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.134)</td></tr><tr id="a0000000219"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y(t),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.135)</td></tr></table><p>
where [mathjaxinline]t[/mathjaxinline] is called a <span style="color:#27408C"><b class="bf">parameter</b></span>. For each real number [mathjaxinline]t[/mathjaxinline], the point [mathjaxinline](x(t), y(t))[/mathjaxinline] is a point on the curve. </p><p><b class="bfseries">Eliminating parameters</b></p><p>
To find the underlying curve, try eliminating the parameter using algebra and/or trig identities. </p><p><b class="bfseries">Tangent lines of parametric curves</b></p><p>
The slope of a parametric curve [mathjaxinline]x=x(t)[/mathjaxinline], [mathjaxinline]y=y(t)[/mathjaxinline] is </p><table id="a0000000220" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\displaystyle \frac{dy}{dx} = \large {\frac{\left(\frac{dy}{dt}\right)}{\left(\frac{dx}{dt}\right)}}.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
In particular, to find the slope of the tangent line to the curve at [mathjaxinline]t=t_0[/mathjaxinline], we compute [mathjaxinline]\displaystyle \frac{y'(t_0)}{x'(t_0)}[/mathjaxinline]. </p><p><b class="bfseries">Arc length of parametric curves</b></p><p>
Consider a particle moving along a trajectory. The motion is described by the parametric curve </p><table id="a0000000221" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000222"><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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.136)</td></tr><tr id="a0000000223"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y(t).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.137)</td></tr></table><p>
The speed of the particle is given by </p><table id="a0000000224" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000225"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{ds}{dt}[/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 {\left(x'(t)\right)^2+\left(y'(t)\right)^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.138)</td></tr></table><p>
The differential arc length element is given by </p><table id="a0000000226" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000227"><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 {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.139)</td></tr></table><p>
Consider the arc length, or distance travelled by the particle from time [mathjaxinline]t_0[/mathjaxinline] to time [mathjaxinline]t_1[/mathjaxinline]. The letter [mathjaxinline]s[/mathjaxinline] is customarily used to denote arc length. You should think of [mathjaxinline]s=s(t)[/mathjaxinline] as a function of time, where [mathjaxinline]s(t)[/mathjaxinline] is the distance travelled by the particle since some starting time. If [mathjaxinline]s_0 = s(t_0)[/mathjaxinline] and [mathjaxinline]s_1=s(t_1)[/mathjaxinline], then the distance traveled by the particle from time [mathjaxinline]t_0[/mathjaxinline] to [mathjaxinline]t_1[/mathjaxinline] can be calculated by </p><table id="a0000000228" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000229"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle s_1 - s_0 = \int _{s_0}^{s_1} ds = \int _{t_0}^{t_1} \sqrt {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.140)</td></tr></table><p><b class="bfseries">A note about notation</b></p><table id="a0000000230" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000231"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle ds^2 = dx^2 + dy^2[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \qquad \text {means} \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle (ds)^2 = (dx)^2 + (dy)^2[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.141)</td></tr><tr id="a0000000232"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle ds = \sqrt {dx^2 + dy^2}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \qquad \text {means} \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle ds =\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">(1.142)</td></tr></table><p>
In particular, [mathjaxinline]\displaystyle dx^2 = (dx)^2.[/mathjaxinline] This is the square of a differential, not the differential of the square. The differential of the square is [mathjaxinline]\displaystyle d(x^2) = 2x\, dx[/mathjaxinline], which is not the same. </p><p><b class="bfseries">Position and speed along a parametric curve</b></p><p>
We have been thinking of a parametric curve as the description of a particle's position over time. So what is the velocity? And what about the acceleration? </p><ul class="itemize"><li><p>
The derivative [mathjaxinline]x'(t)[/mathjaxinline] is the velocity in the direction of the [mathjaxinline]x[/mathjaxinline]-axis. </p></li><li><p>
The derivative [mathjaxinline]y'(t)[/mathjaxinline] is the velocity in the direction of the [mathjaxinline]y[/mathjaxinline]-axis. </p></li><li><p>
The speed along the curve is given by [mathjaxinline]\displaystyle \frac{ds}{dt} = \sqrt {\left(x'(t)\right)^2 + \left(y'(t)\right)^2}.[/mathjaxinline] </p></li><li><p>
The notion of velocity along the curve requires considering [mathjaxinline]x'[/mathjaxinline] and [mathjaxinline]y'[/mathjaxinline] together in what is known as a vector. Both velocity and acceleration are vectors and you will see them in multivariable calculus. </p></li></ul><p><b class="bfseries">Surface area</b></p><p>
Consider the parametric curve </p><table id="a0000000233" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000234"><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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.143)</td></tr><tr id="a0000000235"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y(t).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.144)</td></tr></table><p>
Consider the surface formed by rotating the curve about the [mathjaxinline]y[/mathjaxinline]-axis. </p><div id="a0000000236" class="figure"><center><img src="/assets/courseware/v1/cf7c568d5c7fb89a0f4f2fd708c4aa31/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_arclength_im4.svg" width="500px" alt="See caption" style="margin: 10px 25px 25px 25px"/><div class="caption"><b>Figure 3</b>: <span>Rotating a curve about the [mathjaxinline]y[/mathjaxinline]-axis.<br/></span></div></center></div><p>
The differential surface area element is given by </p><table id="a0000000237" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000238"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle dA[/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 2\pi x\, ds[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.145)</td></tr><tr id="a0000000239"><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 2\pi x(t) \sqrt {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.146)</td></tr></table><p>
The the surface area from time [mathjaxinline]t_0[/mathjaxinline] to time [mathjaxinline]t_1[/mathjaxinline] is given by the integral: </p><table id="a0000000240" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000241"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Surface area of parametric curve} = \int _{t_0}^{t_1} 2\pi x(t) \sqrt {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.147)</td></tr></table><p>
Consider the surface formed by rotating the curve about the [mathjaxinline]x[/mathjaxinline]-axis. </p><div id="a0000000242" class="figure"><center><img src="/assets/courseware/v1/a95843a63a1586dfeff1a87e292f1935/asset-v1:MITx+18.01.3x+1T2020+type@asset+block/images_arclength_im3.svg" width="300px" alt="See caption" style="margin: 10px 25px 25px 25px"/><div class="caption"><b>Figure 4</b>: <span>Rotating a curve about the [mathjaxinline]x[/mathjaxinline]-axis.<br/></span></div></center></div><p>
The differential surface area element is given by </p><table id="a0000000243" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000244"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle dA[/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 2\pi y\, ds[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.148)</td></tr><tr id="a0000000245"><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 2\pi y(t) \sqrt {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.149)</td></tr></table><p>
The the surface area from time [mathjaxinline]t_0[/mathjaxinline] to time [mathjaxinline]t_1[/mathjaxinline] is given by the integral: </p><table id="a0000000246" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000247"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \text {Surface area of parametric curve} = \int _{t_0}^{t_1} 2\pi y(t) \sqrt {\left(x'(t)\right)^2+\left(y'(t)\right)^2}\, dt[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.150)</td></tr></table><p>
The (signed) area under a curve [mathjaxinline]y=f(x)[/mathjaxinline] between [mathjaxinline]x=a[/mathjaxinline] and [mathjaxinline]x=b[/mathjaxinline] is given by [mathjaxinline]\displaystyle \int _ a^ b f(x)\, dx[/mathjaxinline]. </p><p>
Suppose that this curve is parameterized by the equations </p><table id="a0000000248" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000249"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle x[/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 x(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.151)</td></tr><tr id="a0000000250"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y(t).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.152)</td></tr></table><p>
Moreover suppose that [mathjaxinline]x(t_0) = a[/mathjaxinline] and [mathjaxinline]x(t_1) = b[/mathjaxinline]. Then the (signed) area under the curve is also equal to </p><table id="a0000000251" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000252"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _ a^ b f(x)\, 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 \int _{t_0}^{t_1} f(x(t))\, x'(t) \, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.153)</td></tr></table><p>
by change of variables (or by substitution and applying the chain rule). </p><p>
But note that </p><table id="a0000000253" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000254"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle f(x)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.154)</td></tr><tr id="a0000000255"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.155)</td></tr><tr id="a0000000256"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \Longrightarrow y(t)[/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 f(x(t))[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.156)</td></tr></table><p>
hence </p><table id="a0000000257" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000258"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \int _ a^ b f(x)\, 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 \int _{t_0}^{t_1} y(t)\, x'(t) \, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.157)</td></tr></table><p>
This formula for the (signed) area under a parametric curve holds in general! </p><p><b class="bfseries">General result for parametric curves</b></p><p>
Given a parametric curve: </p><table id="a0000000259" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000260"><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:center; border:none">
[mathjaxinline]\displaystyle =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle x(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.158)</td></tr><tr id="a0000000261"><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 =[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle y(t),[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.159)</td></tr></table><p>
The signed area of the region bounded between the curve and the [mathjaxinline]x[/mathjaxinline]-axis for [mathjaxinline]t_0 < t < t_1[/mathjaxinline] is given by the integral </p><table id="a0000000262" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000263"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \textrm{Signed Area}[/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 _{t_0}^{t_1} y(t)\, x'(t) \, dt.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(1.160)</td></tr></table>
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<p>
For a printable version, click <a href="https://courses.edx.org/asset-v1:MITx+18.01.3x+1T2020+type@asset+block@param_summary.pdf" target="_blank">here</a>. </p>
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