Does it work to take samples which are not uniform and then weight each sample differently when computing the integral (such as dividing each sample by the probability of taking the sample at that point)? What advantages are there to doing this if it works?
If we care about samples being uniform, what is the downside of picking a more standard distribution? What benefits do things like circles provide?
How would one uniformly sample from a ball? I'm assuming that this would entail making use of three parameters (a radius, and two angles). One can't pick each of the three parameters uniformly at random, as that results in a similar issue, but I guess one one do something related? (e.g., using equations for the Jacobian of the transformation between a sphere and a prism).
How does taking the square root even the distribution?