Let $I_k:= c \int_{\mathbb R^3} (3x_k'^2-r'^2) \,\,\,d^3 x'$ where ${r'}^2={x'}_1^2+{x'}_2^2+{x'}_3^2$ and $c$ is a constant = density of charge (uniform) in the body.
Suppose this integral is evaluated for a solid spheroid ${x^2\over a^2}+{y^2\over a^2}+{z^2\over b^2}\le 1$
Now suppose we hollow out the spheroid and place all the charge uniformly over the shell. Is there a good way of seeing whether $I_k$ remains constant or changes?