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Given that $f$ is a continuous and increasing function on $[a, b]$, $c = f(a), d = f(b)$ and $a, b, c ,d \geq 0$, explain why

$$\int_c^d f^{-1}(t)~dt = bd - ac - \int_a^b f(x)~dx$$

I am not sure how to treat the inverse function or the functions as the limits of integration. I tried to apply what I found here, but couldn't seem to create the RHS exactly.

$$\int_c^d f^{-1}(t)~dt = F^{-1}(d) - F^{-1}(c)$$ $$= F^{-1}(f(b)) - F^{-1}(f(a))$$ $$ = b - a$$

I am trying to understand exactly what the equality is trying to demonstrate in terms of properties of integrals rather than copying a formula from wikipedia superficially.


So, from the RHS, I would be taking the area of $ab$ (largest rectangle) and subtracting $ca$ (shaded rectangle) and then subtracting the area under $f(t)$ (lightly shaded)?

enter image description here

So I'm finding the area between $f^{-1}$ and the vertical axis from $f(a)$ to $f(b)$.

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Try drawing a picture, just to get a feeling for why this is true. – Dylan Moreland Jun 8 '12 at 6:58
I'm having difficulty drawing it since the functions are abstract. Would it be the area under some increasing inverse function and between two other functions all on the positive $x$ axis? Wouldn't that involve finding the points of intersection and splitting things into 3 integrals? – stariz77 Jun 8 '12 at 7:10
Draw a rectangle, with the x-axis going from a to b, and the y-axis going from c to d. Draw an increasing wiggly line from (a,c) to (b,d). This is $f$. What is $f^{-1}$ on this picture? What are the areas in question? – copper.hat Jun 8 '12 at 7:12
@Ananda: Thanks, momentary mind-blip. I have removed my comment. – copper.hat Jun 8 '12 at 7:22
This seems more like an application of Fubini to me. – copper.hat Jun 8 '12 at 8:11
up vote 5 down vote accepted

For a formal proof, start by substituting $t=f(x)$ in $\int_c^d f^{-1}(t)\,dt$.

For a pictorial proof:

Pictorial proof

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