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<h3 style="margin-top:2ex;padding-top:0;">Name-<br>
Math 213, May 2022, Test 3</h3>
<p>

<p class="justme nexttime answer">First person done at 

<ol>
(12 pts)

<img src="0g/axes.png" class="rightalign" style="width:300px">
Sketch and/or describe the solid whose volume is given by the integral below.  Find the volume either by evaluating the integral or using geometry.
$$V=\int_{r=2}^4\int_{0}^{2\pi}\int_0^2\ r\,dz\,d\theta\,dr.$$




<li>


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<li style="page-break-before: always">
<img src="0g/triple_2022.png" style="width:350px" class="rightalign">
(15 pts) Set up a <i>triple integral</i> to evaluate the volume of the tetrahedron bounded by the coordinate planes and the plane $6x +21y + 14z = 42$, as shown. Then evaluate the
integral. 
</li>


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<li style="page-break-before:always">(6 pts)
<img src="0g/T3.whichvf.png" style="width: 300px" class="rightalign">
Which of the vector fields below describes the plot at right?
<ol type="a">
<li>$x\hat i+(x-y)\hat j$
<li>$y\hat i+(x-y)\hat j$
<li>$x\hat i+(x+y)\hat j$
<li>$y\hat i+(x+y)\hat j$
</ol>
<br clear="both">

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<li style="margin-top: 2ex">
(15 points)
<img src="0g/T1.2.jpg"><br>
<ol type="a">
<li>
What are the coordinates $(\rho, \theta, \phi)$ of $P$ in spherical coordinates?
<p class="justme result">$\rho = \sqrt{10}=3.16$;
<br>$\theta = \arctan(1/1)=\pi/4=0.785$<br>
$\phi = \arcsin(\sqrt 8/\sqrt{10})=\arccos(\sqrt 2/\sqrt{10})=1.11$ [R]=$63.4^o$.

<li>What are the coordinates $(r, \theta, z)$ of $P$ in cylindrical coordinates?
</ol>









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<li style="page-break-before:always">
(18 pts) Consider the integral 
$\int_0^4 \int_{y= x/2}^{\sqrt x} \left(  x+2y \right)\,dy\,dx$.
<ol type="a">
<li>
Sketch the area of integration, labeling all important points.

<li> 
Write out the integral that reverses the order of integration, with appropriate limits.

<li>Evaluate either the original integral, or the one you came up with in part b).  You may use CoCalc or Wolfram Alpha.

</ol>







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<li style="page-break-before:always">
<img src="0g/T3.vfplot.png" class="rightalign" style="margin-right:-4em">
(9 pts) Consider this vector field (at right).<br>
Draw and label 3 curves from (-2,0) to$\to$ (2,0), one for each part below, such that:
<br><br>
a. $\int_A\myv F\cdot d\myv r \gt 0$<br>
b. $\int_B\myv F\cdot d\myv r \lt 0$<br>
c. $\int_C\myv F\cdot d\myv r \approx  0$<br>


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<li style="margin-top:20ex;">
(12 pts) Let $L$ be the directed line segment, $L$ from the point (1,1) to the point (1,3). Given $\myv F=2x\uv i+(x+2y)\uv j$, evaluate the line integral
$\int_L \myv F \cdot d\myv r$.  Show all your work.






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<li style="page-break-before:always">
(15 pts) Let $\myv F(x,y)=(\sin x+ \frac72y^2)\uv i+\left(e^y-\frac 12 x\right)\uv j$. 
<img src="0g/T3.Green.jpg" class="rightalign" style="margin-right:-2em">
Compute the value of the line integral of $\oint_C \myv F\cdot d\myv r$ where
$C$ is the closed path (at right), by means of...
<ol type="a"><li>Converting the line integral to a double integral using Green's theorem (show the resulting double integral),
<li>Then evaluate the double integral.
</ol>

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<li style="margin-top: 50ex">
(15 pts) Let $\myv F(x,y)=(1+2xy)\uv i+(x^2-9y^2)\uv j$.
<ol type="a">
<li>
Use an appropriate test to show that $\myv F$ is a conservative vector field.
<li>
Find a potential function for $\myv F$.
<li>
Let $C$ be the curve $\myv r(t)=(\cos(\pi t) +2)\uv i+(t-\sin(\pi t)\uv j$ with 
$t=0\to 3$.  Based on your answers to a) and b), indirectly compute 
$\int_C\myv F\cdot d\myv r$.
</ol>
</li>

</ol>


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