Finite-thickness hollow-sphere moment of inertia ($C=\frac{2}{5}M\frac{R^5-r_0^5}{R^3-r_0^3}$)

Integrating $dC=(2/3)r^{2}dm$ over a uniform-density spherical layer gives the displayed result. It becomes $2M{R}^2/5$ for a solid sphere and tends to $2M{R}^2/3$ only in the thin-shell limit. Calling a layer hollow does not make its thickness negligible.