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\problemgrouptitle{Integration Applications Review}

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

\begin{problemblock}{Find the area of the region bounded by the given curves.}

\item $\displaystyle y=x^2-2x+4,\,\,\, y=2x-x^2,\,\,x=1,\,\,x=2$  
\answer{
$\displaystyle \frac{8}{3}$
} 

\item $\displaystyle y=x^3,\,\,\, y=x,\,\,y=4x,\,\,y\ge 0$  
\answer{
$\displaystyle \frac {15}{4}$
}

\item $\displaystyle x=y^2-4,\,\,\, x=0$
\answer{
$\displaystyle \frac{32}{3}$
}

\item $\displaystyle y=\cos x,\,\,\, y=\sin x,\,\,x=0,\,\, x=\frac{\pi}2$  
\answer{
$\displaystyle 2\sqrt 2-2$
}

\end{problemblock}

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

\item $\displaystyle y=x^2+2,\,\,\, y=10-x^2;\,\,$ about the $x$-axis
\answer{
$\displaystyle 256\pi$
} 

\item $\displaystyle y=x^2+1,\,\,\, y=3-x^2$ about the line $y=-2$
\answer{
$\displaystyle \frac{64\pi}{3}$
} 

\item $\displaystyle y=4-x^2,\,\,\, x=0,\,\,y=0;\,\,$ about the line $x=-2$
\answer{
$\displaystyle \frac{88\pi}{3}$
} 

\item $\displaystyle y=x^2-2x,\,\,\, y=0;\,\,$ about the $y$-axis
\answer{
$\displaystyle \frac{8\pi}{3}$
} 

\item $\displaystyle y=6-x,\,\,\, y=3x+2,\,\,x=0;\,\,$ about the line $x=8$
\answer{
$\displaystyle \frac{92\pi}{3}$
} 

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

\end{problemblock}

\begin{problemblock}{Find the length of the given curve over the given interval.}

\item $\displaystyle y=(x-1)^{3/2}+2$\ over $1 \le x \le 4$
\answer{
$\displaystyle \frac 1{27}[31^{3/2}-8]$
}

\item $\displaystyle x=\frac 16 y^3+\frac 1{2y}+3$\ over $1 \le x \le 2$
\answer{
$\displaystyle \frac{17}{12}$ 
}

\end{problemblock}

\addtoanswerkey{\end{multicols}}

\item Find the arc length function $s(x)$ of the curve $y=\ln x$ with starting point $P(1,0)$\,.
\answer{
  $\displaystyle s(x)=\ln\left(\frac{\sqrt{1+x^2}-1}{x}\right)+\sqrt{1+x^2}-\ln\left(\sqrt{2}-1\right)-\sqrt{2}$
}

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

\item $\displaystyle y=\sqrt{5-x^2}, \,\,\,\,\,0\le x\le 4$ \ revolved about the $x$-axis
\answer{
$\displaystyle 8\sqrt 5\pi$
}

\item $\displaystyle x=\sqrt{1+3y^2}, \,\,\,\,\,0\le y\le 2$ \ revolved about the $y$-axis
\answer{
$\displaystyle \frac{\pi}{2\sqrt 3}\left[28\sqrt 3+\ln|4\sqrt 3+7|\right]$
}

\end{problemblock}

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