A sphere has a coordinate system (r, $\theta$, $\phi$) with the origin at the center of the sphere.
What is the joint PDF of the r and $\phi$ coordinates, $f_{r,\phi}(r,\phi)$, for a randomly selected point within a region that is defined by two chords that begin at the intersection of the z axis with the sphere and each open from the z axis with an angle of $b$?
The diagram shows a cross section of the region.
I think that to get $f_{r}(r)$ in this problem you would need to find $\frac{d}{dr} \int_{0}^{2\pi}\int_{0}^{r}\int_{\pi-b-\arcsin{\frac{R_a \sin{b}}{\rho}}}^{\pi} \rho^2 \sin{\phi}d\phi d\rho d\theta$ for values of $r$ less that $r_{sphere}$, where I found the limits of integration on the $\phi$ integral by using the law of sines to find $\phi$ in terms of $\rho$, but I'm not sure about how to get $f_{r,\phi}(r,\phi)$ after that.