12

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.

Hosein Rahnama
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Korgan Rivera
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6 Answers6

28

For what it's worth, here's a diagram to accompany Brian M. Scott's and Leandro's answers:

enter image description here

David Mitra
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24

No, but there is a relationship between those two quantities if $f$ is continuously differentiable and strictly increasing (you can relax the last assumption).

By parts, we have $\int_a^b f(x)dx = x f(x)|_a^b - \int_a^b x f'(x)dx = x f(x)|_a^b - \int_a^b f^{-1}(f(x)) f'(x)dx$ = $x f(x)|_a^b - \int_{f(a)}^{f(b)} f^{-1}(u) du $ where in the last step we do the substitution $u=f(x)$.

Chris Janjigian
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  • I think this is the best answer. No need to bring geometry in. Upvoted. – Pedro Feb 07 '12 at 04:52
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    Furthermore, we can get $\int_a^b f(x)dx = x f(x)|a^b - \int_a^b x f'(x)dx = x f(x)|_a^b - \int_a^b f^{-1}(f(x)) f'(x)dx=x f(x)|_a^b - \int{f(a)}^{f(b)} f^{-1}(u) du(u=f(x))=x f(x)|a^b - \int{f(a)}^{f(b)} f^{-1}(x) dx $, so your guess is close to the fact that $\int_a^b f(x) dx = bf(b)-af(a) - \int_{f(a)}^{f(b)}f^{-1}(x)dx$ – Huang Feb 07 '12 at 06:51
  • @Huang You just wrote the same as him. (?) – Pedro Feb 13 '12 at 04:01
  • @Peter Because the op guessed $\int_a^b f(x) dx = (f(b)-f(a))b - \int_{f(a)}^{f(b)}f^{-1}(x)dx$ , I just wan to show him his guess is close to the fact. – Huang Feb 13 '12 at 05:47
  • Also when the function f and its inverse are analytic in a domain about the origin including b, the results are related to the Legendre transformation with f(0)=0 and a=0. – Tom Copeland Apr 07 '12 at 03:26
12

No. Take $f(x) = x^2$, $a = 0$, $b = 1$. Then in this interval $f$ is integrable and invertible, with $f^{-1}(x) = \sqrt{x}$, but the integrals are easily seen to be different.

student
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9

You actually have that:

$$\int_a^b f(x) dx = b f(b) - a f(a) - \int_{f(a)}^{f(b)} f^{-1} (x) dx $$

Here's a graph:

enter image description here

The rectangle $ObCB$ has area $b\cdot f(b)$. The rectangle $OaDA$ has area $a \cdot f(a)$. The curved trapezium $ADCB$ has area $\int_{f(a)}^{f(b)} f^{-1} (x) dx $ so it is expected that:

$$\int_a^b f(x) dx = \mathcal{A}(ObCB) - \mathcal{A}(OaDA) - \mathcal{A}(ADCB)$$ $$\int_a^b f(x) dx = b f(b) - a f(a) - \int_{f(a)}^{f(b)} f^{-1} (x) dx $$

which is actually true.

Remeber that the starting function has to be one-to-one and onto for the inverse to be defined.

You can check a full proof in Michael Spivak's Calculus (though he want's you to do it, he provides the steps necessary to do so).

Pedro
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4

No.

Suppose that $f(x)$ is a continuous, strictly increasing function on $[a,b]$. Then $\int_a^b f(x)dx$ gives the area between the curve and that segment $[a,b]$ on the $x$-axis, while $\int_{f(a)}^{f(b)}f^{-1}(x)dx$ gives the area between the curve and the segment $[f(a),f(b)]$ on the $y$-axis, and it’s easy to see that these two areas need not be the same. (There are easy concrete examples $-$ I see now that Leandro has given one $-$ but even a few pictures should convince you.)

Brian M. Scott
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-1

∫baf(x)dx + ∫f(b)f(a)f−1(x)dx = bf(b)-af(a) is the expression to be used. The proof of the same can be seen here

Prove that $\int_a^bf(x)dx+ \int_{f(a)}^{f(b)} f^{-1}(y)dy=bf(b)-af(a)$

SchrodingerCat2023
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    As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community May 19 '23 at 02:58