I've become confused reading over some notes that I've received. The integral in question is \int_{|x| < \sqrt{R}} |x|^2\, dx where $x \in \mathbb{R}^n$ and $R > 0$ is some positive constant. The notes state that because of the spherical symmetry of the integrand this integral is the same as $\omega_n \int_0^{\sqrt{R}} r^2 r^{n-1}\, dr$. Now neither $\omega_n$ nor $r$ are defined. Presumably $r = |x|$, but I am at a loss as to what $\omega_n$ is (is it related maybe to the volume or surface area of the sphere?).
I am supposing that the factor $r^{n-1}$ comes from something like $n-1$ successive changes to polar coordinates, but I am unable to fill in the details and would greatly appreciate any help someone could offer in deciphering this explanation.