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{\bf Exam Question
Topic: TripleIntegral}
A solid circular cylinder has radius 2, and the distance between
its circular ends is 6. The density at a point $P$ of the cylinder
is proportional to the product of the square of the distance of
$P$ from the axis of the cylinder and the distance of $P$ from the
nearest circular end of the cylinder> Find the total mass, and the
average density of the cylinder. \vspace{0.5in}
{\bf Solution} In
cylindrical polars
\begin{eqnarray*}
M&=&2k\int_0^{2\pi}\, d\phi\int_0^3\, dz\int_0^2 r^2z.r\, dr =
4\pi k\int_0^3 z\, dz\int_0^2 r^3\, dr\\ &=&4\pi
k\,\frac{9}{2}\frac{16}{4}=72\pi k.
\end{eqnarray*}
the volume of the cylinder is $\pi.2^2.6=24\pi.$
So the average density is $\displaystyle{\frac{72\pi
k}{24\pi}}=3k.$
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