= Cartesian second moment of a solid ellipsoid
{title2=$\int_Dx^2\,dV=4\pi a^3bc/15$}
For the <solid ellipsoid> $D=\{x^2/a^2+y^2/b^2+z^2/c^2\le1\}$ with positive semi-axes, set $x=au$, $y=bv$, $z=cw$. The <Jacobian determinant> of this inverse coordinate map is $abc$, and $D$ becomes the unit ball. Symmetry gives $\int_{B_1}u^2\,dV=\frac13\int_{B_1}r^2\,dV=4\pi/15$, proving the displayed moment. The other two moments follow by interchanging the semi-axes; mixed moments vanish by reflection symmetry.
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