Solution
= Solution
For the <tangent-axis inertia of a spherical shell>, subtract the missing solid core from a solid sphere:
$$
M=\frac{4\pi\rho}{3}(R_2^3-R_1^3),\qquad
I_c=\frac{8\pi\rho}{15}(R_2^5-R_1^5).
$$
The <parallel axis theorem> shifts the axis a distance $R_2$:
$$
\boxed{I_{\rm tangent}=\frac{8\pi\rho}{15}(R_2^5-R_1^5)
+\frac{4\pi\rho}{3}(R_2^3-R_1^3)R_2^2.}
$$
It tends to $7MR_2^2/5$ for a solid sphere and $5MR_2^2/3$ for a thin <spherical shell>.