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\centerline{MATH 136 -- Calculus 2}
\centerline{Practice Day on Volumes of Solids of Revolution}
\centerline{March 11, 2020}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
As we have seen in some examples, let the region $R$ be defined by 
$0 \le y \le f(x)$ for all $x$ in $[a,b]$.
Then the volume of the solid of revolution obtained by rotating $R$ about
the $x$-axis is given by
\begin{equation}
\label{volbasic}
\text{V (circle cross-sections) = \int_a^b \pi (f(x))^2\ dx
\end{equation}
since the cross-section at $x$ is a circle with radius $r = f(x)$.  
Note that this region ``extends all the way to the $x$-axis'' (the axis of rotation).
If there is a ``gap'' between the region and the axis of rotation, then the cross-sections
will be ``washers'' and we need to set up the volume integral differently:
\begin{equation}
\label{volwasher}
\text{V (washer cross-sections)} = \int_a^b \pi (\text{outer radius})^2 - (\text{inner radius})^2\ dx
\end{equation}
Today we want to practice on using \eqref{volbasic} and \eqref{volwasher} to set up and compute
integrals for volumes of some regions of this type.
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\noindent
{\it Questions}
\vskip 10pt
For each problem, 
\begin{enumerate}
\item[(i)] Set up the volume integral for all of the following regions first. 
\item[(ii)] The go back and evaluate each integral using the FTC part I.  You 
may use a table of integrals anywhere here, as needed.  
\end{enumerate}
\begin{enumerate}
\item  The solid generated by rotating the region between $y = x^4$
and the $x$-axis, $[a,b] = [0,1]$.
\item  The solid generated by rotating the region between $y = x^4$ and $y = x$, $[a,b] = [0,1]$.
\item  The solid generated by rotating the region between $y = \sec(x)$, 
and $y = \cos(x)$,  $[a,b] = [0,\pi/4]$.
\item  The solid generated by rotating the region between 
$y = \cdot (1 - x^2)^{1/4}$ and the $x$-axis, $[a,b] = [0,1]$.  
\item  $f(x) = \sin(x)$, $[a,b] = [0,\pi]$.  
\end{enumerate}
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\begin{figure}
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\includegraphics[width=.8\linewidth]{needleregion.png}
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\caption{The region for problem 4.}
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