Here is a proof of integration by parts: http://www.math.ufl.edu/~mathguy/year/S10/int_by_parts.pdf
But I don't understand how it works. Specifically, I don't understand why $\int_r^s u \, dv$ equals one of the areas (and likewise for $\int_p^q v \, du$).
To me it looks like this only works if $g$ is the inverse of $f$...
Thanks!