Is this right?(something i discovered )

[Andrax]

Is this right?

Homework Statement

Hello , so in spivak's calculus we only learn to calculate the integral of x^2 and x in the first chapter , i was wonderin gwhat would the integral of $x^(1/2)$ $\int_a^b$x^(1/2)$~dx$ , well i used the inverse function and a little bit of work i calculated from 0 to 2 using the integral of x^2 on 0 to 2^(1/2) and integral of 2^(1/2) on 0 to 2 turned out it's the correct answer using wolfram alpha , i checked other variables same , so i tried to generalize this , i don't know if this is correct but here it is (f-1 = f^-1 inverse function of f , it wouldn't let me write it on latex)
$\int_a^b f(x)~dx$=-$\int_c^d f^-1(x)~dx$+$\int_a^b f(b)-f(a)~dx$
where c = f^-1(a) and d=f^-1(b)
i didn't check for a lot of functions , but i was wondering if this is right ; thanks everyone :).

 

[micromass]

Close, but I don't think it's entirely correct. The correct formula is in Spivak, Exercise 22 in Chapter 13. It holds only for increasing $f$ and states

$$\int_a^b f^{-1} = bf^{-1}(b) - af^{-1}(a) -\int_{f^{-1}(a)}^{f^{-1}(b)} f$$

I don't think your formula is quite equivalent to this.

 

[Andrax]

Close, but I don't think it's entirely correct. The correct formula is in Spivak, Exercise 22 in Chapter 13. It holds only for increasing $f$ and states

$$\int_a^b f^{-1} = bf^{-1}(b) - af^{-1}(a) -\int_{f^{-1}(a)}^{f^{-1}(b)} f$$

I don't think your formula is quite equivalent to this.

oh so there is aformula for it , well thank you