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%  Areas of Surface of Revolution                  %
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\problemgrouptitle{Areas of Surface of Revolution}

\begin{problemblock}{Find the area of the surfaces of revolution generated by the given curves.}

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

\item $\displaystyle y=2x+5, \,\,\,\,\,0\le x\le 2$ \ revolved about the $x$-axis
\answer{
$\displaystyle 28\sqrt 5\pi$
}

\item $\displaystyle y=(9-x^{2/3})^{3/2}, \,\,\,\,\,1\le x\le 8$ \ revolved about the $x$-axis
\answer{
$\displaystyle \frac {18\pi} 5 [8^{5/2}-5^{5/2}]$
}

\item $\displaystyle y=\frac 1{20}x^5+\frac 13x^{-3}, \,\,\,\,\,1\le x\le 2$ \ revolved about the $x$-axis
\answer{
$\displaystyle \frac{4907\pi}{1600}$ 
}

\item $\displaystyle y=\sqrt{6-x}, \,\,\,\,\,1\le x\le 2$ \ revolved about the $x$-axis
\answer{
$\displaystyle \frac {\pi}6 [21^{3/2}-17^{3/2}]$
}

\item $\displaystyle x=\frac 16 y^3, \,\,\,\,\,0\le y\le 2$ \ revolved about the $y$-axis
\answer{
$\displaystyle \frac 29 \pi [5^{3/2}-1]$
}

\item $\displaystyle x=\sqrt{y-y^2}, \,\,\,\,\,0\le y\le 2$ \ revolved about the $y$-axis
\answer{
$\displaystyle 2\pi$
}

\item $\displaystyle x=\frac 13 (y^2+2)^{3/2}, \,\,\,\,\,0\le y\le 1$ \ revolved about the $x$-axis
\answer{
$\displaystyle \frac{3\pi}2$
}

\item $\displaystyle y=\frac 13 x^3, \,\,\,\,\,1\le x\le 3$ \ revolved about the $y$-axis
\answer{
$\displaystyle \frac{ \pi}2 \left[\sqrt{82}+\ln\left|9+\frac{\sqrt{82}}{9}\right|-\sqrt2-\ln\left|1+\sqrt 2\right|\right]$
}

\addtoanswerkey{\end{multicols}}

\end{problemblock}

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