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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 9/17/97}\\
{\it Closed book \hfill Van Dommelen \hfill 10:45-11:35am}
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Show all reasoning and intermediate results leading to your answer.
One book of mathematical tables, such as Schaum's Mathematical Handbook,
may be used.

\begin{enumerate}

\item You have been given an elliptically-shaped outer contour for a
new building.  You want to put in a rectangular room of the largest
area that you possible can.  So, if the ellipse has a semi-major axis
$a$ and a semi-minor axis $b$,  what is the area of the largest
rectangle that will fit inside?
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<A HREF="figures/1.gif">Solution.</A>
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\item Draw the curve $xy = e^x$.  Identify the $x$- and $y$-extents,
the asymptotes, intercepts, and behavior at large $|x|$.
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<A HREF="figures/2.gif">Solution.</A>
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\item A plate is bounded by the unit circle, the line $x=y$ and
the positive $x$-axis.  The thickness of the plate is $xy$,
so that the mass is given by $\int\int x y\/ {\rm d}x{\rm d}y$.
Find this integral over the given region.
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<A HREF="figures/3.gif">Solution.</A>
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\end{enumerate}

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