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How to use this function to evaluate this integral

  1. Feb 4, 2012 #1
    Hey guys! I'm not from an English speaking country so I'll do my best to translate. I would appreciate if you could help me with this problem.

    1. The problem statement, all variables and given/known data

    Increasing (growing) function is f(0)=0 and f(a)=b and a,b>0
    Use ∫(from 0 to a)f(x)dx=ab-∫(from 0 to b)f^-1(x)dx to solve ∫(from 0 to 2)arcsin(x/4)


    3. The attempt at a solution

    I know how to find ∫(from 0 to 2)arcsin(x/4) so that's no problem but I just don't understand how I use the other part to find it, what am I supposed to do? Help would be appreciated!

    Thanks in advance!
     
  2. jcsd
  3. Feb 4, 2012 #2

    HallsofIvy

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    Your formula says that
    [tex]\int_0^a f(x)dx= ab- \int_0^b f^{-1}(x)dx[/tex]

    Here, [itex]f(x)= arcsin(x/4)[/itex] so what is [itex]f^{-1}(x)[/itex]? a= 2 and b= arcsin(1/2). What is that?
     
  4. Feb 4, 2012 #3
    Thanks for the reply! I understand now how to think this but I still get wrong answer. So a=2 and therefore b must be arcsin(2/4) like you said. f(x)^-1 is sin(x/4).

    When I put all this into the function I get 2arcsin(1/2) - ∫(from 0 to arcsin(1/2))sin(x/4) = 1.01....

    But when I calculate ∫(from 0 to 2)arcsin(x/4) I get 0.51....

    What am I doing wrong?

    EDIT: OK never mind, I got the right answer. Thanks for your help!
     
    Last edited: Feb 4, 2012
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