# Integrating a spherically symmetric function over an $n$-dimensional sphere

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.

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The $\omega_n$ is meant to be the volume of the unit $n$-sphere. See http://en.wikipedia.org/wiki/N-sphere for notation.
Also, you are correct that $r = |x|$, and the $r^{n-1}$ comes from the Jacobian of the transformation from rectangular to spherical coordinates. (The $\omega_n$ also comes from this transformation).