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%  Determining the Volume Method                   %
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\problemgrouptitle{Determining the Volume Method}

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

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

\item $\displaystyle y=x^2,\,\,\, y=2-x\, ;\,\,$ about the line $y=-1$
\answer{
$\displaystyle  \frac{117\pi}{5}$
}  

\item $\displaystyle y= -x^2+6x,\,\,\, y=x^2-2x;\,\,$ about the line $x=5$
\answer{
$\displaystyle 128\pi$
} 

\item $\displaystyle y=x^2,\,\,\, y=-x^2+8\,;\,\,$ about the line $y=-1$
\answer{
$\displaystyle \frac{640\pi}{3}$
} 

\item $\displaystyle y=x^2,\,\,\, y=2x+3;\,\, $ about the line $y=-1$
\answer{
$\displaystyle \frac{1408\pi}{15}$
} 

\item $\displaystyle y=-x^2+4,\,\,\, y=-3x;\,\,$ about the line $x=-2$
\answer{
$\displaystyle \frac{875\pi}{6}$
} 

\item $\displaystyle x=-y^2+8,\,\,\, x=2y ;\,\,$ about the line $y=4$
\answer{
$\displaystyle  \frac{1080\pi}{3}$
} 

\item $\displaystyle x=-y^2,\,\,\, x=-2y^2+4;\,\,$ about the line $x=6$
\answer{
$\displaystyle \frac{1664\pi}{15}$
} 

\item $\displaystyle y=x^3,\,\,\, y=\sqrt{x};\,\,$ about the line $y=-1$
\answer{
$\displaystyle \frac {25\pi}{21}$
} 

\item $\displaystyle y=4x-x^2,\,\,\, y=0;\,\,$ about the $y$-axis
\answer{
$\displaystyle \frac{128\pi}{3}$
}

\item $\displaystyle y= x^2-2x+4,\,\,\, y=2x^2-3x+2;\,\,$ about the line $x=4$ 
\answer{
$\displaystyle \frac{61\pi}{2}$
}

\addtoanswerkey{\end{multicols}}

\end{problemblock}

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