Question: Let $h$ be a nonnegative measurable function on $\mathbb{R^3}$. Rewrite $$\int_{\mathbb{R^4}}\int_{\mathbb{R^4}}\ h\left(|x|^2,|y|^2,x\cdot y\right)\ dx\ dy$$ as $$\iiint_{\Omega} \ J(u,v,w)\ h(u,v,w)\ du\ dv\ dw$$ where $\Omega$ is an appropriate domain in $\mathbb{R^3}$ and $J(u,v,w)$ is an appropriate function on $\Omega$.
My Thoughts: I'm assuming that this has to deal with Fubini/Tonelli theorems. I suppose I am just unsure on how to deal with $h$, and then how to construct $J$ from that $h$. I know that the area of the unit $k$-dimensional sphere in $\mathbb{R^{k+1}}$ is $w_k$, where $w_1=2\pi$, $w_2=4\pi$, and $w_3=2\pi^2$. So, am I suppose to be taking a $3$-dimensional unit sphere in $\mathbb{R^4}$, and then trying to rewrite $h$ on that sphere, and use the measure of that sphere to come up with our $\Omega$? Any thoughts, ideas, etc. are very much appreciated! Thank you.