Radial integration over a ball (${\int }_{\Vert x\Vert \le R}\varphi (\Vert x\Vert )dx=d{V}_d{\int }_0^R\varphi (r)r^{d-1}dr$)

Let $V_d$ be the volume of an n-ball of unit radius in ${\mathbb{R}}^d$. A radial shell has volume $d{V}_d{r}^{d-1}dr$, by differentiating $V_d{r}^d$. Integrating shells, or using hyperspherical coordinates, therefore reduces a radial integrand over a Euclidean ball to the displayed one-dimensional integral. The constant $d{V}_d$ is the surface area of an n-sphere of unit radius, with sphere dimension $d-1$.