Tangent-axis inertia of a spherical shell ($I=\frac{8\pi \rho }{15}(R_2^5-R_1^5)+M{R}_2^2$)

For uniform density between concentric radii $R_1,R_2$, subtract the missing solid core's central moment of inertia and then shift the axis by $R_2$. Here $M=4\pi \rho (R_2^3-R_1^3)/3$. The thin-shell limit gives $I=5M{R}_2^2/3$, while a full solid sphere gives $7M{R}_2^2/5$.