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Math Help - Application of Green's Theorem

  1. #1
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    Application of Green's Theorem

    Hi, I'm trying to answer the following question:



    Basically, I know that you have to use Green's Theorem:

    \displaystyle\oint_C P(x)dx +Q(x)dy = \displaystyle\iint_A (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\ dx\,dy

    Which makes our integral:

    \displaystyle\iint_A x \ dx\,dy

    My problem is, I don't know how to calculate the limits for the area A. Do we use a parameterisation, or are the limits simply the points of intersection of the functions.

    Any help is immensely appreciated.

    Thanks
    Last edited by Ackbeet; May 21st 2011 at 02:36 AM. Reason: Fixed LaTeX code to use [tex] delimiters.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    The intersection points of y=x and y=x^2-2x are (0,0) and (3,3) . You'll obtain:


    \displaystyle\oint_C 3xy\;dx +2x^2\;dy=\iint_Ax\;dxdy= \int_0^3\;dx\int_{x^2-2x}^xx\;dy=\ldots
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  3. #3
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    IF you were to integrate around the boundary, you would need to parameterize the curve. You don't use parametric equations for an area in 2 dimensions. That's why it is easier to use Green's theorem.
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