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\begin{center}
{\Large TEST 3

Math 2210 - 2}
\end{center}

Instructions:
\begin{itemize}
\item  You have 50 minutes to complete this exam.
\item  Show all your work.  If I can't follow your work or it is missing you WILL NOT get credit.  
	(This holds for graphing too!)
\item  Use the back of these sheets if you run out of room.  Make sure I can find your work.
\item  No calculators or notes.
\item  The test is out of 40 points -- each problem is worth 10 points.
\end{itemize}

\hrule

\begin{enumerate}

\item  Consider the rectangle $R = \{(x,y) \stbar 0 \leq x \leq 1, -1 \leq y \leq 1 \}$
	and compute the integral:
	\[
	\dint_R xy^3 \; dA
	\]


\item  Find the maximum and minimum of $f(x,y,z)=x-y+z$ subject to the constraint:
\[
x^2+y^2+z^2 = 2
\]



\item  Let $T$ be the region in the plane bounded by the curves:
\[
y=0 \qquad x=2 \qquad y=\ln x
\]
Consider the integral
\[
\dint_T x \; dA
\]
Set this integral up as an iterated integral in two ways by finding the appropriate limits of integration
\begin{enumerate}
\item  Find the limits of integration for
	\[
	\int\int x \; dx \; dy
	\]
\item  Find the limits of integration for
	\[
	\int\int x \; dy \; dx
	\]
\item  Pick one of your integrals above and compute it.
\end{enumerate}




\item  Let $R$ be the region in the plane defined by
	\[
	R = \{ (x,y) \stbar 1 \leq x^2+y^2 \leq 4, x \geq 0, y \geq 0 \}
	\]
 Compute the integral
\[
\dint_R \frac{1}{\sqrt{x^2+y^2}} \; dA
\]











\end{enumerate}




\end{document}







