Chapter 14.8, Problem 54ES

The tendency of a solid to resist a change rotational motion about an axis is measured by its moment of inertia about that axis. If the solid occupies a region G in an xyz-coordinate system, and if its density function $\delta \left(x,y,z\right)$ is continuous on G, then the moment of inertia about x-axis, the y-axis and the z-axis are denoted by $I_x,I_y\text{and}{I}_z,$ respectively, and are defined by

$\begin{array}{l}{I}_x={\displaystyle \underset{G}{\iiint }\left(y^2+z^2\right)\delta \left(x,y,z\right)dV}\\ I_y={\displaystyle \underset{G}{\iiint }\left(x^2+z^2\right)\delta \left(x,y,z\right)dV}\\ I_z={\displaystyle \underset{G}{\iiint }\left(x^2+y^2\right)\delta \left(x,y,z\right)dV}\end{array}$

In these exercises, find the indicated moments of inertia of the solid, assuming that it has constant density $\delta .$

$I_z$ for the solid sphere $x^2+y^2+z^2\le a^2.$