"If the improper integral for surface area converges if and only if the improper integral for volume converges! This might actually be true which I am going to try to prove."
Greetings PerfectHacker Sir:
May I assume your assertion stated in blue above to correctly read: "The improper integral for surface area converges if and only if the improper integral for volume converges" --irrespective of truth value, of course? That is, is the opening word "If" a type-O? If that be the case, what are your thoughts regarding the convergent integral y=e^-x, x in [0,inf) and bound by, say, z=0 and z=1 for instance? Here we have finite volume bound by infinite surface area. Does this example not serve to counter your assertion? Or have I failed to comprehend some aspect of the claim?
Season's Greetings to you.