MATH 136 – Calculus 2 Discussion Day 4 – Areas and Volumes March 17, 2014

Group write-ups of solutions for the following problems are due by class time on Wednesday, March 19.

I. Let R be the region bounded by y = xe^{−x}, y = 0, x = 0 and x = 4.

A) Sketch the region. B) A solid has the region R as base and cross-sections perpendicular to the x-axis that

are squares (extending the full width of the base). Find the volume. C) Set up and evaluate the integral to find the volume of the solid obtained if R is rotated

about the x-axis. D) Set up and evaluate the integral to find the volume of the solid obtained if R is rotated

about the line y = −2.

II. The hull of a boat is 20 feet long. At a distance s feet from the bow (the front), the cross section of the hull has the shape of the region in the plane below y = 0 (y = 0 is the water line) and above the parabola y = ax^{2} − b, where a, b are given in the following table:

\begin{matrix}s & 5 & 10 & 15 & 20 \\ a & 2 & 3 & 4 & 5 \\ b & 2 & 3 & 4 & 4 \end{matrix}

Estimate the volume enclosed by the hull below the waterline y = 0.