Density-power cylindrical moment of inertia
= Density-power cylindrical moment of inertia
{title2=$I_z=MR^2(p+2)/(p+4)$}
For a solid cylinder with <mass density> $\rho(r)=kr^p$ and $p>-2$, integrating cylindrical shells gives $M=2\pi LkR^{p+2}/(p+2)$ and axial <moment of inertia> $I_z=2\pi LkR^{p+4}/(p+4)$. Their ratio proves the displayed formula. It gives the uniform-cylinder result $MR^2/2$ at p=0 and $2MR^2/3$ at p=2. The radial density weighting must enter both mass and inertia integrals.