= Radial integration over a ball
{title2=$\int_{\|x\|\le R}\phi(\|x\|)\,dx=dV_d\int_0^R\phi(r)r^{d-1}\,dr$}
Let $V_d$ be the <volume of an n-ball> of unit radius in $\mathbb R^d$. A radial shell has volume $dV_dr^{d-1}dr$, by differentiating $V_dr^d$. Integrating shells, or using <hyperspherical coordinates>, therefore reduces a radial integrand over a <Euclidean ball> to the displayed one-dimensional integral. The constant $dV_d$ is the <surface area of an n-sphere> of unit radius, with sphere dimension $d-1$.
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