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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.

  • 2
    You may be interested in [Young's inequality for increasing functions](http://en.wikipedia.org/wiki/Young%27s_inequality#Standard_version_for_increasing_functions).2012-02-07

5 Answers 5