Axial moment of inertia of a hollow cylinder
= Axial moment of inertia of a hollow cylinder
{title2=$I=\tfrac12 M(R_o^2+R_i^2)$}
For a cylinder of uniform <mass density> $\rho$, length $L$, inner radius $R_i$ and outer radius $R_o$, an annular element has $dm=2\pi\rho Lr\,dr$. Thus the axial <moment of inertia> is $I=\int r^2dm=\pi\rho L(R_o^4-R_i^4)/2$. Dividing by $M=\pi\rho L(R_o^2-R_i^2)$ yields the displayed result. At fixed <mass>, the solid-cylinder limit is $MR_o^2/2$ and the thin-shell limit is $MR^2$.