Finding the Mass of a Sphere

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 \[ \begin{eqnarray*} M &=&\iiint\limits_{E}\delta \,{dV}=\iiint\limits_{E}k\rho \rho ^{2}\sin \phi \,d\rho \,d\phi \,d\theta =k\iiint\limits_{E}\rho ^{3}\sin \phi \,d\rho \,d\phi \,d\theta\\ &=&k\int_{0}^{2\pi }\int_{0}^{\pi }\left[ \int_{0}^{a}\rho ^{3}\,d\rho \right] \sin \phi \,d\phi \,d\theta \notag \\ & =&k\int_{0}^{2\pi}\int_{0}^{\pi }\left[ \dfrac{\rho ^{4}}{4}\right] _{0}^{a}\,\sin \phi \,d\phi \,d\theta =\dfrac{ka^{4}}{4}\int_{0}^{2\pi }\left[ \int_{0}^{\pi }\sin \phi \,d\phi \right] d\theta\\ &=&\dfrac{ka^{4}}{4} \int_{0}^{2\pi }\big[ -\cos \phi \big] _{0}^{\pi }\,d\theta \notag \\ & =&\dfrac{ka^{4}}{2}\int_{0}^{2\pi }\,d\theta =k\pi a^{4} \end{eqnarray*} \]