I haven't read through all this lengthy thread, but I know it can be done; I did it myself many years ago and have been trying to remember the details of the algebra involved. My approach involved imagining a hemisphere with n horizontal disks of equal thickness inscribed therein.
I haven't...
Is this because what the OP's really wants to prove is the volume of the closed ball comprising the sphere and its interior?
Forgive what's undoubtedly a simple question, but this isn't my field. I came in because I was interested in the same proof (or, at least, a proof of the same fact)...