Find the mass $M$ of a sphere if its mass density $\delta$ is proportional to the distance from the center of the sphere.

Solution Let $a$ be the radius of the sphere, and position the sphere so that its center is at the origin. In spherical coordinates, the equation of the sphere is $\rho =a$. The mass density $\delta$ of the sphere is $\delta =k\rho,$ where $k$ is the constant of proportionality. The mass $M$ is $$\begin{eqnarray*} M &=&\iiint\limits_{E}\delta \,{dV}=\iiint\limits_{E}k\rho \rho ^{2}\sin \phi \,d\rho \,d\phi \,d\theta =k\iiint\limits_{E}\rho ^{3}\sin \phi \,d\rho \,d\phi \,d\theta\\ &=&k\int_{0}^{2\pi }\int_{0}^{\pi }\left[ \int_{0}^{a}\rho ^{3}\,d\rho \right] \sin \phi \,d\phi \,d\theta \notag \\ & =&k\int_{0}^{2\pi}\int_{0}^{\pi }\left[ \dfrac{\rho ^{4}}{4}\right] _{0}^{a}\,\sin \phi \,d\phi \,d\theta =\dfrac{ka^{4}}{4}\int_{0}^{2\pi }\left[ \int_{0}^{\pi }\sin \phi \,d\phi \right] d\theta\\ &=&\dfrac{ka^{4}}{4} \int_{0}^{2\pi }\big[ -\cos \phi \big] _{0}^{\pi }\,d\theta \notag \\ & =&\dfrac{ka^{4}}{2}\int_{0}^{2\pi }\,d\theta =k\pi a^{4} \end{eqnarray*}$$