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Math Help - Reversing Green's Theorem

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    Reversing Green's Theorem

    Given that R is the region bounded by the curve C defined by 4x^2+9y^2=36, then by using Green's Theorem, the double integral int int x^2 dA is equal to...?

    Any idea to solve it?
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    Quote Originally Posted by prescott2006 View Post
    Given that R is the region bounded by the curve C defined by 4x^2+9y^2=36, then by using Green's Theorem, the double integral int int x^2 dA is equal to...?

    Any idea to solve it?
    Green's theorem states that

    \int_{C} Pdx+Qdy=\int \int_D \left( \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right) dA

    so to use the theorem backwards we need to find a function that P or Q that would gives the proper integrand.

    There is more than one, for example

    let P=0 \mbox{ and } Q=\frac{1}{3}x^3

    Then by greens theorem we get

    \int_C \frac{1}{3}x^3dy=\int \int_Dx^2dA

    note that simple closed path is r(t)=3\cos(t) \vec i +2\sin(t) \vec j \mbox{ for } 0 \le t \le 2\pi
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