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\begin{center}
{\Large TEST 3

Math 2210}
\end{center}

\begin{itemize}
\item  You have 1 hour to complete this exam.
\item  Show all your work (and make it neat!).  
  If I can't follow your work or it is missing you WILL NOT get credit.
\item  You can use a $4 \times 6$ note card of notes.
\item  If you run out of space, use the back of these sheets.
\end{itemize}


\vspace{.4in}

Name: \hrulefill


\begin{enumerate}


\item  Compute the triple integral below
\[
\iiint_E \sqrt{x^2+y^2} \; dV
\]
Where $E$ is the solid bounded by the paraboloid $z=9-x^2-y^2$ and the $xy$-plane.


\item  Let $S$ be the region in space defined by the inequalities:
\[
z \geq 0 \qquad x \geq z \qquad x \leq -2z+4 \qquad y \leq 5 \qquad  y \geq z
\]
\begin{enumerate}
\item  Write a triple integral that represents the volume of $S$.
\item  Evaluate your triple integral to compute the volume of $S$.
\end{enumerate}



\item  Compute the following line integral:
\[
\int_C (x^2+y)\;dx + (y^2+x)\; dy
\]
where $C$ is the curve $y=x^2$ from $(3,9)$ to $(-2,4)$.



\item  Consider the line integral below:
\[
\int_C y \; dx + (x+z^2)\; dy + 2yz\; dz
\]
where $C$ is a curve from $(1,2,3)$ to $(-1,-1,2)$.
\begin{enumerate}
\item  Find a vector field $\mathbf{F}$ so that the line integral can be written $\int_C F \dotprod d\mathbf{r}$.
\item  Show that the above integral is independant of path.
\item  Find a potential function for $\mathbf{F}$.
  \\
  Hint: Notice that $\mathbf{F}$ should be a vector in $\R^3$ and the potential function should be a function of three variables.

\item  Using the potential function, compute the line integral.
\end{enumerate}





\end{enumerate}

\end{document}







