Chapter 15.6, Problem 38E

Moments of Inertia In Exercises 37-40, use the following formulas for the moments of inertia about the coordinate axes of a surface lamina of density $\rho$.

$\begin{array}{cc}{I}_x={\displaystyle {\int }_S{\displaystyle \int \left(y^2+z^2\right)\rho \left(x,y,z\right)dS}}& I_y={\displaystyle {\int }_S{\displaystyle \int \left(x^2+z^2\right)\rho \left(x,y,z\right)dS}}\\ I_z={\displaystyle {\int }_S{\displaystyle \int \left(x^2+y^2\right)\rho \left(x,y,z\right)ds}}& \end{array}$

Verify that the moment of inertia of a spherical shell of uniform density about its diameter is $\frac{2}{3}m{a}^2$, where m is the mass and a is the radius.