= Isotropic second moment of a ball
{title2=$\int_{|x|\le R}x_ix_j\,d^dx=\frac{V_dR^{d+2}}{d+2}\delta_{ij}$}
Here $V_d$ is the <volume> of the unit ball in $d$-dimensional <Euclidean space>. Rotational symmetry makes the integral an <isotropic second-rank tensor> $C\delta_{ij}$. Taking its <trace> gives $dC=\int_{|x|\le R}|x|^2\,d^dx=dV_d\int_0^Rr^{d+1}dr$. This proves the formula. For a constant <matrix> $M$, contraction gives $\int x^TMx\,d^dx=V_dR^{d+2}\operatorname{tr}M/(d+2)$. In three dimensions the coefficient is $4\pi R^5/15$.
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