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Is this right?(something i discovered )

  1. Jul 31, 2013 #1
    Is this right?

    1. The problem statement, all variables and given/known data
    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 :).
     
    Last edited: Jul 31, 2013
  2. jcsd
  3. Jul 31, 2013 #2

    micromass

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    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

    [tex]\int_a^b f^{-1} = bf^{-1}(b) - af^{-1}(a) -\int_{f^{-1}(a)}^{f^{-1}(b)} f[/tex]

    I don't think your formula is quite equivalent to this.
     
  4. Jul 31, 2013 #3
    oh so there is aformula for it , well thank you
     
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