This is an exercise from Folland: Suppose $f(x)=\prod_{j=1}^n x_j^{\alpha_j}$ where all $\alpha_j$ are even. Then show that $$\int_{\mathbb{S}^{n-1}} f d \sigma= \frac{2 \prod_{j=1}^{n}\Gamma(\beta_j)}{\Gamma\left(\sum_{j=1}^n\beta_j\right)} \quad \text{where }\beta_j=\frac{\alpha_j+1}{2}$$.
As hint Folland suggests to calculate $\displaystyle \int_{\mathbb{R}^n} e^{-|x|^2}f(x)\mathrm{d}x$.
Attempt: By Tonelli's we have
$$\int_{\mathbb{R}^n} e^{-|x|^2}f(x)\mathrm{d}x=\prod_{j=1}^{n}\int_{-\infty}^{\infty}e^{-y^2}y^{\alpha_j} \mathrm{d}y=\prod_{j=1}^{n}\Gamma(\beta_j)$$
I think I am supposed to change to polar coordinates now $(r=|x|, \, x'=\frac{x}{|x|})$. Since $f(rx')=r^{\sum \alpha_j}f(x')$, we get:
$$\prod_{j=1}^{n}\Gamma(\beta_j)=\int_{\mathbb{R}^n} e^{-|x|^2}f(x)\mathrm{d}x=\int_0^{\infty}e^{-r^2}r^{(\sum \alpha_j)+n-1}\mathrm{d}r \int_{\mathbb{S}^{n-1}} f \mathrm{d}\sigma$$ from which the result follows.
Is this correct?