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      <text>27.4.4.4 Volume of a Torus</text>
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      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>A torus (doughnut) has major radius \(R\) and minor radius \(r\). Using the transformation </p>
<p> \(\qquad \displaystyle x=(R + \tau \cos \alpha) \cos \theta \, \), \(y=(R + \tau \cos \alpha) \sin \theta \, \), \(z=\tau \sin \alpha \, \), </p>
<p> find the volume of the torus, shown in the figure below. </p>
<p> [Hint 1: the limits on \(\alpha\) and \(\theta\) are \(0\) to \(2\pi\), and the limits on \(\tau\) are \(0\) to \(r\).] </p>
<p> [Hint 2: the Jacobian is \(\tau(R + \tau \cos \alpha)\)]

&nbsp;</p><p><img src="@@PLUGINFILE@@/27-4-figure-torus.svg" alt="The described torus" width="396" height="198" class="img-fluid atto_image_button_text-bottom"><br></p>

<p> Volume \(= \) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p>
<p> [Note: leave your answer in terms of \(\pi, R, \, \text{and} \, r\), which can be entered as <code>pi</code>, <code>R</code>, and <code>r</code> respectively.]</p>]]></text>
<file name="27-4-figure-torus.svg" path="/" encoding="base64">[encoded data, 10576 characters]</file>
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    <generalfeedback format="html">
      <text><![CDATA[<p> We are asked to find the volume of the torus with major radius \(R\) and minor radius \(r\) using the transformation</p>
<p> \(\qquad \displaystyle x=(R + \tau \cos \alpha) \cos \theta \, \), \(y=(R + \tau \cos \alpha) \sin \theta \, \), \(z=\tau \sin \alpha \, \). </p>
<p> We are told that the limits on \(\alpha\) and \(\theta\) are \(0\) to \(2\pi\), and the limits on \(\tau\) are \(0\) to \(r\), and additionally we are given that the Jacobian is \(\tau(R + \tau \cos \alpha)\).</p>

<p> Since in general, the volume of a shape is given by </p>
<p> \(\qquad \displaystyle \text{Volume} = \int \int \int \mathrm{d}V,\) </p>
<p> and we know that the volume element \(\mathrm{d}V=\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\) becomes \(\mathrm{d}V=|J(\tau,\alpha,\theta)|\; \mathrm{d}\tau\,\mathrm{d}\alpha\,\mathrm{d}\theta\), we have everything we need to find the volume of the torus. </p>

<p> \( \begin{align} 
\text{Volume} &= \int_{0}^{2\pi} \int_{0}^{2\pi} \int_{0}^{r} \tau(R + \tau \cos \alpha) \, \mathrm{d}\tau \,\mathrm{d}\alpha \,\mathrm{d}\theta \\ 
&= \int_{0}^{2\pi} \int_{0}^{2\pi} \int_{0}^{r} \left(\tau R + \tau^2 \cos \alpha\right) \, \mathrm{d}\tau \,\mathrm{d}\alpha \,\mathrm{d}\theta \\ 
&= \int_{0}^{2\pi} \int_{0}^{2\pi} \left[\dfrac{1}{2} \tau^2 R + \dfrac{1}{3} \tau^3 \cos \alpha \right]_{0}^{r} \,\mathrm{d}\alpha \,\mathrm{d}\theta \\ 
&= \int_{0}^{2\pi} \int_{0}^{2\pi} \left(\dfrac{1}{2} r^2 R + \dfrac{1}{3} r^3 \cos \alpha \right) \,\mathrm{d}\alpha \,\mathrm{d}\theta \\ 
&= \int_{0}^{2\pi} \left[\dfrac{1}{2} r^2 R \alpha + \dfrac{1}{3} r^3 \sin \alpha \right]_{0}^{2\pi}  \,\mathrm{d}\theta \\ 
&= \int_{0}^{2\pi} \left(\dfrac{1}{2} r^2 R (2\pi) + \dfrac{1}{3} r^3 \sin (2\pi) - 0\right) \,\mathrm{d}\theta \\ 
&= \int_{0}^{2\pi} \pi r^2 R \,\mathrm{d}\theta \\ 
&= \displaystyle \left[\pi r^2 R \theta\right]_0^{2\pi} \\
&= 2 \pi^2 R r^2. \end{align}\).</p>]]></text>
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      <text>ta1: 2*pi^2*R*r^2;</text>
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      <text>Volume = {@ta1@}</text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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