EML 5060 Analysis in Mechanical Engineering 9/17/97
Closed book Van Dommelen 10:45-11:35am

Show all reasoning and intermediate results leading to your answer. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used.

1.

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? Solution.

2.

Draw the curve xy = e^x. Identify the x- and y-extents, the asymptotes, intercepts, and behavior at large |x|. Solution.

3.

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. Solution.