Average over a sphere surface

 The surface integral of the function $f$ over the surface $S$ is denoted by

\[ \int\int_S f dS \]

$dS$ is the area of an infinitesimal piece of the surface $S$.

Average over the sphere is

\[ <f> = \frac{1}{4\pi} \int_0^{2\pi} d\phi \int_0^{\pi} f \sin(\theta)d\theta ,\]

where $dS=\sin(\theta)d\theta d\phi$. This can be seen as a weight average, the weight is surface area.

\[ <\cos^2(\theta)>=1/3 \]

\[<\sin(\theta)>=2/3.\]