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<h3 style="margin-top:2ex;padding-top:0;">Name-<br>
Math 213, May 2018, Test 3</h3>
<p>All question parts are worth 5 pts, unless otherwise noted. (Total 105).
<ol>




<li>
Consider the triple integral 
$\int_0^1 \int_{y^3}^{\sqrt y}\int_0^{xy}dz\,dx\,dy$ which represents the volume of a solid $S$. Let $R$ be the projection of $S$ onto the plane $z=0$.
<ol type="a">
<li>
Carefully draw the region $R$ on the $xy$ plane, taking care to label all important features.

<li>
Rewrite the integral as 
$\iiint_S dz\,dy\,dx$ (different order of integration), but don't evaluate the integral.
</ol>
</li>

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<li style="margin-top: 34ex">
<img src="0g/T3.tetrahedron.jpg" class="rightalign">
(10 pts) Set up a <i>triple integral</i> to evaluate the volume of the tetrahedron bounded by the coordinate planes and the plane $10x +15y + 6z = 30$, as shown. Then evaluate the
integral. 
</li>

<!---3. Pac Man ------------------>
<li style="page-break-before:always">
<img src="0g/T3.pacman.jpg" class="rightalign">
(10 pts) Consider the region $P$ in the shape of Pac-Man.
Find $\iint_P x\,dA$ by converting the integral to <i>polar coordinates</i> then solving the integral.






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<li style="page-break-before:always">
(10 pts) Let $\myv F(x,y)=2y\uv i+\uv j$. Carefully sketch several representative vectors at the following 7 points: (0,0), (1,0), (-1,0), (0,1), (0,-1), (1,-1), (-1,1).<br>
<img src="0g/T3.grid.jpg"><br>

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<li style="margin-top:5ex">
<img src="0g/T3.divfield.jpg" class="rightalign" style="margin-right:-4em">
Let $\myv F$ be the vector field sketched in the diagram.
<ol type="a">
<li>
Where is the curl of $\myv F$ negative / zero / positive?

<li style="margin-top: 10ex">
Where is the divergence of $\myv F$ negative / zero / positive?

<li style="margin-top: 10ex">
Is $\myv F$ a conservative field? Why or why not?
</ol>


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<li style="page-break-before:always">
Let $\myv F(x,y)=(3+2xy)\uv i+(x^2-3y^2)\uv j$.
<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 $\myv r(t)=(-\cos t -1)\uv i+(-\sin t)\uv j$ with 
$0\leq t\leq \pi$.  Based on your answers to a) and b), indirectly compute 
$\int_C\myv F\cdot d\myv r$.
</ol>
</li>

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<li style="margin-top:45ex">
Let $\myv F(x,y,z)=xy\uv i+(y^2-2z)\uv j+\sin(yz)\uv k$.
<ol type="a">
<li>
Compute the divergence of $\myv F$.

<li>
Compute the curl of $\myv F$.

</ol>



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<li style="page-break-before:always">
<img src="0g/T3.lineintegrals.jpg" class="rightalign" style="margin-right:-4em">
Write the three line integrals<br>
$\int_C\myv F\cdot d\myv r$, $\int_D\myv F\cdot d\myv r$, $\int_E\myv F\cdot d\myv r$<br>
in order of increasing value:<br><br>
$\ \ \int_\text{___}\myv F\cdot d\myv r \lt \int_\text{___}\myv F\cdot d\myv r \lt \int_\text{___}\myv F\cdot d\myv r$<br>




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<li style="margin-top:20ex;">
(10 pts) Let $L$ be the directed line segment from the point (1,4,2) to the point (3,4,1). Given $\myv F=2x\uv i+(y^2+z)\uv j+(x+y)\uv k$, 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">
Let $\myv F(x,y)=(\sin x+ xy^2)\uv i+\left(e^y+\frac 12 x^2\right)\uv j$. 
<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 evaluating the double integral.
</ol>

</ol>


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