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Integration is the 'area under the curve' for a one dimensional function. Say we have a function [tex]f(x)=x^2[/tex], if we compute the integral of this function between two limits we get the area under it between the two limits on the x axis.

What if we wanted to find the area between two limits on the 'f(x) axis'?

The usual way (i.e. the way I've been taught) would be define a new function, say g(x) which is the inverse of the function f(x) and calculate the integral between the desired limits for g(x).

I was thinking of 'swapping' x and f(x) in the integrand. We just need to make sure the limits are right and hey presto. I've tried it so far for a couple of functions and it seems to work out.

It looks like this:

[tex]\int_a^b f(x) dx[/tex]

We now want the integral on the y-axis for the same limits.

So swap x with f(x).

[tex]\int_{f(a)}^{f(b)} x df(x)[/tex] = [tex]\int_{f(a)}^{f(b)} x \frac{d}{dx}f(x) dx[/tex]

Is this right? The whole reason for doing it this way is you don't have to find an inverse function, even though its not that difficult to do in this case.

I imagine that not all inverses are defined for an arbitrary region of interest, so the whole process of finding an inverse is more general than this method.

A quick counterexample that I can think of would be trying to find the integral this way of f(x)= sin(x) from 0 to [tex]4\pi[/tex].