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{\bf Exam Question

Topic: TripleIntegral}

A hemispherical solid of radius $a$ is resting on its circular
face on a horizontal plane. The density at any point of the solid
is proportional to the square of the distance of the point from
the horizontal plane. Calculate the mass of the object and its
mean density. \vspace{0.5in}

{\bf Solution}

Density $=kz^2$ in cartesian coordinates. We perform the
integration in spherical polars.

\begin{eqnarray*}
M &=& \int_0^a\, dr\int_0^{2\pi}\,
d\phi\int_0^{\pi/2}kr^2\cos^2\theta r^2\sin\theta\, d\theta \\ &=&
k\int_0^a r^4\, dr\int_0^{2\pi}\, d\phi
\int_0^{\pi/2}\cos^2\theta\sin\theta\, d\theta \\ &=&
\frac{ka^5}{5}\, 2\pi\,
\left[-\frac{\cos^3\theta}{3}\right]_0^{\pi/2} = \frac{2\pi
k^5}{15}
\end{eqnarray*}

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