= Cartesian moments of a three-dimensional ball
{c}
{title2=$\int_{|x|\leq a}x_ix_j\,dV=\frac{4\pi a^5}{15}\delta_{ij}$}
<Spherical symmetry> makes moments over a ball <isotropic tensors>. Odd moments vanish under $x\mapsto-x$, while the second and fourth moments are
$$
\int_{|x|\leq a}x_ix_j\,dV=\frac{4\pi a^5}{15}\delta_{ij},\qquad
\int_{|x|\leq a}x_ix_jx_kx_l\,dV=\frac{4\pi a^7}{105}(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}).
$$
For the second moment, a <tensor contraction> gives $3\alpha=\int r^2\,dV=4\pi a^5/5$. For the fourth, complete index symmetry makes the three coefficients equal and a double <tensor contraction> gives $15\beta=\int r^4\,dV=4\pi a^7/7$. These identities turn polynomial <volume integrals> into scalar radial integrals and are useful for inertia and rotational averages.
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