Uniform Distribution on a Sphere: Part 3b (The Solution by Calculus)

(This post continues where we left off in Part 2 and gives a more direct solution than Part 3a. However this post uses multivariable/vector calculus, and I do not provide many explanations for readers unfamiliar with that material.)

As we saw in Part 2, the problem with picking our spherical angles $\theta$ and $\phi$ uniformly was that the area on our two-dimensional grid changed when we transformed that surface into the sphere in three dimensions. In particular if we wanted to calculate the probability mass in a region $R$ of our spherical angles, we have the formula from Part 0:

where $f_{\theta,\phi}$ is our probability density function. However when we move to the sphere our area elements are no longer little rectangles with area $\mathrm{d}\theta\cdot\mathrm{d}\phi$; they are little patches on the surface of a sphere. What area do these corresponding patches have?