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{\bf Exam Question

Topic: TripleIntegral}

Let $C$ denote a solid cylinder of height 2 and radius 1 whose
axis of symmetry is the $z$-axis. The density of this cylinder at
a point $P$ is equal to the product of the distance of $P$ from
the bottom of the cylinder and the distance of $P$ from the
$z$-axis. Find the total mass of the cylinder $C$ by evaluating an
appropriate triple integral. \vspace{0.5in}

{\bf Solution}

In cylindrical polar coordinates the density is $rz.$ So the mass
is given by
$$\int_{\phi=0}^{\pi}\,d\phi\int_{z=0}^2\,dz\int_{r=0}^1rz.r\, dr
=2\pi\int_0^2 z\, dz\int_0^1 r^2\, dr
=2\pi\times2\times\frac{1}{3}=\frac{4\pi}{3}$$


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