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{\bf Exam Question

Topic: Volume of Revolution}

Find the volume of revolution obtained by rotating the region in
the $x$-$y$ plane bounded by the lines $x=0,y=0,x=1$ and the curve
$y=\mathrm{e}^x$ about (i) the line $y=3$, (ii) the line $x=-2$.

Give your answer in terms of e, and also as an approximation
correct to 3 decimal places, using your calculator.
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{\bf Solution}


(i)
\begin{eqnarray*}
V &=&\pi\int_0^1[3^2-(3-\mathrm{e}^x)^2]\,
dx=\pi\int_0^1(6\mathrm{e}^x-\mathrm{e}^{2x})\, dx\\ &=&
\pi\left[6\mathrm{e}^x-\mathrm{e}^{2x}/2\right]_0^1 =
\pi[6\mathrm{e}-\mathrm{e}^2/2-6+1/2]\\&=&
\frac{\pi}{2}[12\mathrm{e}+\mathrm{e}^2-11]=22.353\ \mathrm{(3\
d.p.)}
\end{eqnarray*}

(ii)
\begin{eqnarray*}
V &=&2\pi\int_0^1(x+2)\mathrm{e}^x\, dx
=\left[2\pi(x+2)\mathrm{e}^x\right]_0^1-\int_0^12\pi\mathrm{e}^x\,
dx \\ &=&
\left[2\pi(x+2)\mathrm{e}^x\right]_0^1-\left[2\pi\mathrm{e}^x\right]_0^1
= 2\pi(2\mathrm{e}-1)=27.867\ \mathrm{(3\ d.p.)}
\end{eqnarray*}




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