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Math Help - Evaluating definite integral

  1. #1
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    Evaluating definite integral

    I know the value of the following definite integral

    \int_{a}^{b}ydx

    I also have a realtion

    x=f(y)

    i.e. x is an explicit function of y but I do not have y as an explicit
    function of x. The relation between x and y is generally non linear.

    Now I want to get the following definite integral

    \int_{a}^{b}\left[\int ydx\right]xdx

    i.e. \int ydx multiplied by x evaluated over the interval [a,b].

    Is there an analytic (not numeric) way to evaluate this integral using
    for example mean value or similar averaging technique?
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  2. #2
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    Re: Evaluating definite integral

    Can you actually state the problem you have instead of using generalities?
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  3. #3
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    Re: Evaluating definite integral

    There is no generality apart from x=f(y).
    So assume
    x=1+y-3y^3-9sin(y)
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  4. #4
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    Re: Evaluating definite integral

    To find \int y dx you can first get the derivative of f(y)

    \frac{dx}{dy}=f'(y)

    Rearrange for dx

    dx=dy f'(y)

    Then change the variable in the integral

     \int y (f'(y) dy)

    After that do integration by parts. By letting dv= f'(y) dy, you can see simply that v=f(y)

    After the integration by parts suppose that the answer you get is g(y)

    Then to find

    \int_a^b g(y) x dx

    Do the same trick as before to change the dx to dy, and substitute f(y) for x. And also change the limits of the integral to the new variable. So you get

    \int_{f^{-1}(a)}^{f^{-1}(b)} g(y) f(y) f'(y) dy

    Since you dont have y as an explicit function of x you will have to find f^{-1}(a) by numerical methods
    Last edited by Shakarri; June 30th 2014 at 05:31 AM.
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  5. #5
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    Re: Evaluating definite integral

    Quote Originally Posted by JulieK View Post
    I know the value of the following definite integral

    \int_{a}^{b}ydx

    I also have a realtion

    x=f(y)

    i.e. x is an explicit function of y but I do not have y as an explicit
    function of x. The relation between x and y is generally non linear.

    Now I want to get the following definite integral

    \int_{a}^{b}\left[\int ydx\right]xdx
    The way you have written this makes no sense. You need different variables for the two integrals. One way to write it would be \in_a^b\int_0^x tf^{-1}(t)dt dx.
    If f is not invertible that integral is not defined.

    i.e. \int ydx multiplied by x evaluated over the interval [a,b].

    Is there an analytic (not numeric) way to evaluate this integral using
    for example mean value or similar averaging technique?
    Last edited by HallsofIvy; July 1st 2014 at 07:06 AM.
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