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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?

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    As a rule "output values" change when the "input" is changed. If in a particular setup they don't we have a "theorem". - In the case at hand the question would heavily profit from a thorough editing.2012-10-22
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    Thank you, @ChristianBlatter . I am not entirely sure what you mean...2012-10-22

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