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\centerline{MATH 136 -- Calculus 2}
\centerline{Discussion Day 4 -- Areas and Volumes}
\centerline{March 17, 2014}
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\noindent
Group write-ups of solutions for the following problems are due
by class time on Wednesday, March 19.
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\noindent
I.  Let $R$ be the region bounded by $y = xe^{-x}$, $y = 0$,
$x = 0$ and $x = 4$.  
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\item{A)}  Sketch the region.
\item{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.
\item{C)}  Set up and evaluate
the integral to find the volume of the solid obtained if $R$ is rotated
about the $x$-axis.
\item{D)}  Set up and evaluate the integral to
find the volume of the solid obtained if $R$
is rotated about the line $y = -2$.

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\noindent
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:
$$\matrix{s & 5 & 10 & 15 & 20 \cr
          a & 2 & 3 & 4 & 5 \cr
          b & 2 & 3 & 4 & 4 \cr}$$
Estimate the {\it volume\/} enclosed by the hull below the waterline
$y = 0$.
\bye
