<?php $classpath="/213/"; include "../../classes/1.h"; ?>
<title>
Have a great summer!
</title>
<meta name="week" content="">
<!--meta name="nolink" content="nolink"-->
<?php include "../../classes/2.h" ?>
<style type="text/css">
.answer {color:blue;}
</style>

<h3 style="margin-top:2ex;padding-top:0;">Name-<br>
Math 213, May 2020, Test 3</h3>
<p>This is an open book exam.  You may consult your notes / class notes. You may use CoCalc / GeoGebra software as calculational aids.  But you may not consult other people.
<p>For the purposes of partial credit, show plenty of calculations. Add extra blank pages if you need to.

<ol>




<li>
<img src="0g/T3.tetrahedron.jpg" class="rightalign">
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>

<!--------------------->
<li style="page-break-before:always">
Compute the divergence of the 2-dimensional vector field
$\myv F(x,y)=(-\sin(2x)+6y)\uv i+(\ln(x+y)+xy^2)\uv j$.






<li style="margin-top:48ex">
Compute the line-integral, $\int_P f(x,y) ds$ <img src="0g/test3path.jpg" class="rightalign" style="width:350px">of $f(x,y)=3x+\frac 1y$ along the path, $P$, which runs as shown, from point $A$ to $B$ to $C$.<br>






<!--------------------->
<li style="page-break-before:always">
<img src="0g/T3.whichvf.png" style="width: 300px" class="rightalign">
Which of the vector fields below describes the plot at right? (The lengths of the vectors have been shortened by a scale factor.)
<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">





<!---3. Pac Man ------------------>
<li style="margin-top: 5ex">
<img src="0g/T3.pacman.jpg" class="rightalign">
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.








<!--------------------->
<li style="page-break-before:always">
<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: 8ex">
Where is the divergence of $\myv F$ negative / zero / positive?

<li style="margin-top: 8ex">
Is $\myv F$ a conservative field? Why or why not?
</ol>


<!--------------------->
<li style="margin-top: 15ex">
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 
$t=0\to\pi/2$.  Based on your answers to a) and b),  compute 
$\int_C\myv F\cdot d\myv r$.
</ol>
</li>




<!--------------------->
<li style="page-break-before:always">
<img src="0g/T3.vfplot.png" class="rightalign" style="margin-right:-4em">
Consider this vector field (at right).<br>
Draw and label 3 curves, $A$, $B$, and $C$, from (-2,0) to (2,0) 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>


<!--------------------->
<li style="margin-top:20ex;">
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.









<!--------------------->
<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 evaluate the double integral.
</ol>

</ol>

<!--------------------->
<p style="page-break-before:always">
Have a good summer!  

<?php include "../../classes/3.h" ?>
