Density-power cylindrical moment of inertia ($I_z=M{R}^2(p+2)/(p+4)$)

For a solid cylinder with mass density $\rho (r)=k{r}^p$ and $p>-2$, integrating cylindrical shells gives $M=2\pi Lk{R}^{p+2}/(p+2)$ and axial moment of inertia $I_z=2\pi Lk{R}^{p+4}/(p+4)$. Their ratio proves the displayed formula. It gives the uniform-cylinder result $M{R}^2/2$ at p=0 and $2M{R}^2/3$ at p=2. The radial density weighting must enter both mass and inertia integrals.