I completely understand how to transform Lebesgue integration to integration by polar-coordinates using the surface measure.
However, i wonder if there is a weaker version of this justifying integration by polar-coordinates for Riemann-integration.
I don't remember it exactly, but i remember that i have learned somewhat similar result which justifies polar-integration.
Is there a weak theorem for justifying polar-integration to which is strong enough to be applied Riemann integration?
You can derive a formula for coordinate changes (which includes the change to polar coordinates) with the help of the change of variables formula and a version of Fubinis theorem, for which you can give a proof for the Riemann integral. Spivak does this in his 'Calculus on manifolds' book.
The approach is unpleasant though, since the Riemann integral is difficult to handle, and in more than one variable it is even worse. I recall that a lecturer followed this approach in his lecture (some 25 years ago ;-) and he, afterwards, said that he won't ever try this again...