Question 9 (Unit 17) - 13 marks
A thick spherical shell occupies the region between two spheres of radii \( a \) and \( 2 a \), both centred on the origin. The shell is made of a material with density \( \rho=\frac{A z^{2}}{\sqrt{x^{2}+y^{2}+z^{2}}} \), where \( A \) is a constant.
(a) Show that the density expressed in spherical coordinates \( (r, \theta, \phi) \) is
\[
\rho=\operatorname{Ar} \cos ^{2} \theta .
\]
(b) Hence, or otherwise, find the mass \( M \) of the shell by evaluating a suitable volume integral.
(Hint: To evaluate the integral
\[
\int_{0}^{\pi} \sin \theta \cos ^{2} \theta d \theta
\]
use the substitution \( u=\cos \theta \).)
