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%  Calculation of Volume                           %
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\problemgrouptitle{Calculation of Volume}

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

\addtoanswerkey{\begin{multicols}{2}}
  
\item $\displaystyle y=\sqrt x,\,\,\, x=1,\,\,x=9;\,\,$ about the $x$-axis
\answer{
$\displaystyle  40\pi$
} 

\item $\displaystyle y=(x-2)^2,\,\,x=0,\,\,x=2;\,\,$ about the $x$-axis
\answer{
$\displaystyle \frac{32\pi}{5}$
} 

\item $\displaystyle x=y^2+4,\,\,\, y=0,\,\,y=2;\,\,$ about the $y$-axis
\answer{
$\displaystyle \frac{896\pi}{15}$
} 

\item $\displaystyle y=\sqrt{x-4},\,\,\,x=4,\,\,x=8\,\,$ about the $x$-axis
\answer{
$\displaystyle  8\pi$
} 

\item $\displaystyle y=x^{1/3},\,\,\, x=0,\,\,x=8;\,\,$ about the $x$-axis
\answer{
$\displaystyle \frac{96\pi}{5}$
} 

\item $\displaystyle y=1+x^2,\,\,\, y=x,\,\,x=0,\,\,x=2;\,\,$ about the $x$-axis
\answer{
$\displaystyle \frac {166\pi}{15}$
} 

\item $\displaystyle x=y^2,\,\,\, x=2y+3;\,\,$ about the $y$-axis
\answer{
$\displaystyle \frac{1088\pi}{15}$
} 

\item $\displaystyle y=\sqrt x,\,\,\, y=\frac {1}{2}\, x;\,\,$ about the $y$-axis
\answer{
$\displaystyle \frac{64\pi}{15}$
} 

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

\item $\displaystyle  \,\, y=x^3,\,\,\, y=x,\,y\ge 0;\,\,$ about the line $y=2$
\answer{
$\displaystyle  \frac {17\pi}{21}$
} 

\addtoanswerkey{\end{multicols}}

\end{problemblock}

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