I have been constructing some examples related to spherical harmonics. I wanted to know the $L^2$ norm of the function $f(x,y,z) \in S^2 = \{ x^2 + y^2 + z^2 = 1\}$. Wikipedia suggests this normalization:
$$ \int_{S^2} (xy)^2\, dS = \frac{4\pi}{15}$$
Are there any derivations that do not involve spherical coordinates? I am hoping for a more algebraic derivation. Maybe it will use spherical coordinates implicitly.
\begin{eqnarray*} x &=& \cos \theta \cos \phi \\ y &=& \sin \theta \cos \phi \\ z &=& \sin \phi \end{eqnarray*}
For smaller degree terms there is a trick. I would do the integral of $1$:
$$ \lvert\lvert\,1 \,\rvert\rvert^2 = \frac{1}{4\pi}\int_{S^2} 1 \, dS = 1$$
The averages of homogenous linear terms (or cubic terms) is zero. Quadratics have a trick:
$$ \lvert\lvert\,x \,\rvert\rvert^2 = \lvert\lvert\,y \,\rvert\rvert^2 = \lvert\lvert\,z \,\rvert\rvert^2 = \frac{1}{4\pi}\int_{S^2} x^2 \, dS = \frac{1}{3} \frac{1}{4\pi}\int_{S^2} (x^2 + y^2 + z^2) \, dS = \frac{1}{3}$$
I don't think there's such a trick for quadratic polynomials of 4th degree
Through spherical coordinates or not, $$ I(k)= \iint_{S^2}(xy)^{2k}\,d\mu $$ boils down to a value of Euler's Beta function. We have $$ \iint_{x^2+y^2=R^2}(xy)^{2k}\,d\mu=R^{4k+1}\int_{0}^{2\pi}\left(\cos\theta\sin\theta\right)^{2k}\,d\theta =2\pi R^{4k+1}\cdot\frac{1}{4^{2k}}\binom{2k}{k}$$ hence $I(k)$ can be computed from Cavalieri's principle. Or
$$\begin{eqnarray*} I(k)&=&\int_{0}^{2\pi}\int_{0}^{\pi}\sin(\theta)^{4k+1}\cos(\varphi)^{2k}\sin(\varphi)^{2k}\,d\theta\,d\varphi\\&=&2\sqrt{\pi}\frac{\Gamma\left(k+\tfrac{1}{2}\right)^2}{\Gamma\left(2k+\tfrac{3}{2}\right)}=\color{red}{\frac{4\pi\binom{2k}{k}}{(4k+1)\binom{4k}{2k}}}.\end{eqnarray*}$$