\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Start with the region $A$ in the
first quadrant enclosed by the $x$-axis and the parabola
$y=2x(2-x)$. Then obtain solids of revolution $S_1$, $S_2$, and $S_3$
by revolving $A$ about the lines $y=4$, $y=-2$, and $x=4$
respectively. All three solids are (unusual) ``doughnuts'' which are
$8$ units across, whose hole is $4$ units across, and whose height is
$2$ units. Sketch them.

\medskip

\noindent a) Which do you expect to have larger volume, $S_1$ or
$S_2$? Compute their volumes exactly and check your guess.

\medskip

\noindent b) Compute the volume of $S_3$. (It may be harder to guess
in advance how $S_3$ compares in volume to $S_2$ and $S_1$.)

\vfil\eject\end