Is it true that
$$\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$$
Just making sure.
If not, how about:
$$\int_a^b f(x) dx = (f(b)-f(a))b - \int_{f(a)}^{f(b)}f^{-1}(x)dx$$
I'm having a hard time concentrating right now, and I'm trying to figure out how to get the area under a curve when the function is inverted.