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Definite Integrals with inverse of function

  1. Apr 17, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose f(x) is continuous and decreasing on the closed interval 5 <= x < 13, that f(5) = 9, f(13) = 5 and that the

    integral of f(x) from 5 to 13 is 70.64758.

    Then the integral of f^-1(x) from 5 to 9 is equal to what?

    (Note: f^-1(x) is the inverse of f(x))


    2. Relevant equations




    3. The attempt at a solution

    I really don't know how to solve this problem. I know that f^-1(f(x)) = x. Any ideas would be great. Thanks.
     
  2. jcsd
  3. Apr 17, 2009 #2
    You want to find [tex]\int_{u=5}^9 f^{-1}(u)\,du[/tex]


    Try the substitution u=f(x)
     
  4. Apr 17, 2009 #3
    f is a positive monotone function on [a,b] where 0 < a < b and f has an inverse [tex] f^-1 [/tex]. Set [tex] \alpha = f(a), \beta = f(b) [/tex] and then use this formula:[tex] \int_{\alpha}^{\beta} f^-1(y) dy = b \beta - a \alpha - \int_a^b f(x) dx [/tex].
     
  5. Apr 17, 2009 #4
    This formula can be seen by drawing an example graph. Along the x-axis, you are integrating f(x)dx. ALong the y-axis you are integrating f-1(y)dy. Draw some rectangles and add/subtract areas to get the formula.
     
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