Answer to Question #212195 in Physics for shadr

Question

 Derive the formula for the moment of inertia of a uniform sphere of radius r0 and mass M about an axis through its center.[Hint: Divide thesphere into infinitesimally thin disks of thickness of dy and integrate over these cylindrical disks.]

Answer 1

$$dI=0.5r^2dm\\dI=0.5r^2\rho dV\\dI=0.5r^2\rho \pi r^2dx\\ r^2=R^2-x^2\\dI=0.5\rho \pi (R^2-x^2)^2dx\\I=\frac{16}{30}\rho \pi R^5 \\I=\frac{2}{5} M R^2$$