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%  The Shell Method                                %
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\problemgrouptitle{The Shell Method}

\begin{problemblock}{Find the volume of the solid obtained by revolving the given region about the given axis.}

\addtoanswerkey{\begin{multicols}{2}}

\item $\displaystyle y=8-x^2,\,\,\, y=8-3x\, ;\,\,$ about the $y$-axis
\answer{
$\displaystyle \frac{27\pi}{2}$
} 

\item $\displaystyle y=4-x^2,\,\,\, y=0,\,\,x=0\, ;\,\,$ about the $x$-axis
\answer{
$\displaystyle \frac{256\pi}{15}$
} 

\item $\displaystyle y=x^{1/3},\,\,\, y=x^2\,;\,\,$ about the $y$-axis
\answer{
$\displaystyle \frac{5\pi}{14}$
}  

\item $\displaystyle y=x^{1/2},\,\,\, y=x^2 ;\,\,$ about the $x$-axis
\answer{
$\displaystyle \frac{3\pi}{10}$
} 

\item $\displaystyle y+x=6,\,\,\, y=x,\,\,y=1 ;\,\,$ about the $x$-axis
\answer{
$\displaystyle \frac{40\pi}{3}$
} 

\item $\displaystyle y=x^3,\,\,\, y=2x^2 ;\,\,$ about the line $x=3$
\answer{
$\displaystyle \frac{24\pi}{5}$
} 

\item $\displaystyle x=y^2,\,\,\, x-y=2;\,\,$ about the line $y=4$
\answer{
$\displaystyle  \frac {9\pi}{2}$
} 

\item $\displaystyle y= x^3+2,\,\,\, y=4-x^2,\,\,x=2;\,\,$ about the line $x=3$
\answer{
$\displaystyle \frac {53\pi}{5}$
} 

\item $\displaystyle y=x^2,\,\,\, y=2x+3;\,\,$ about the line $x=4$
\answer{
$\displaystyle  {64\pi}$
} 

\item $\displaystyle y=2 \sqrt{x},\,\,\, y=x;\,\,$ about the line $y=-1$
\answer{
$\displaystyle  {16\pi}$
} 

\addtoanswerkey{\end{multicols}}

\end{problemblock}

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