= Second vertical moment of a solid ring torus
{title2=$\int_V z^2\,dV=\pi^2AR^4/2$}
A <solid torus> of major radius $A$ and minor radius $R<A$, centred on the horizontal plane, has volume $2\pi^2AR^2$. In <circular torus coordinates>, $z=r\cos\alpha$. Integrating $z^2$ with the <Jacobian determinant> $r(A+r\sin\alpha)$ gives $\int_Vz^2dV=(\pi^2/2)AR^4$; the part odd in $\sin\alpha$ vanishes. Thus the mean squared height is $R^2/4$. The identity is useful for <moments of inertia> and <flux> calculations using the <divergence theorem>.
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