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Math Help - help with greens theorem

  1. #1
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    help with greens theorem

    just trying figure this example out using greens theorem:


    <br />
I = \oint_R {9y^3 dx - 9x^3 dy} <br />


    mapped counter-clockwise, is the circle

    <br />
x^2  + y^2  = 81<br />

    enter as a multiple of pi

    so using substitution,

    <br />
\int {udx + vdy = \int_{} {\int_R {(\frac{{\delta y}}<br />
{{\delta x}} - \frac{{\delta u}}<br />
{{\delta x}}} } } )dxdy<br />

    so

    <br />
u = 9y^3 <br />
    <br />
v =  - 9x^3 <br />

    and

    <br />
 - 27\int {\int_R {x^2  + y^2 dxdy} } <br />

    <br />
 - 27\int\limits_0^{2\pi } {\int\limits_0^1 {r^2 rdrd\theta } } <br />

    so i got this far how does this end bit work?? do i integrate? where do i go from here...
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  2. #2
    Eater of Worlds
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    Doesn't r vary from 0 to 9?.

    -27\int_{0}^{2\pi}\int_{0}^{9}r^{3}drd{\theta}
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  3. #3
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    oops yeah my mistake, so how would i proceed, from here, i know im supposed to change polars, but i dont know exactly how to solve this
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  4. #4
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    Just integrate. You should know how to do this simple integration since you're in Calc III.

    \int_{0}^{9}r^{3}dr=\frac{1}{4}r^{4}

    \frac{1}{4}(9)^{4}-\frac{1}{4}(0)^{4}=\frac{6561}{4}

    Now, integrate wrt to theta:

    \int_{0}^{2\pi}\frac{6561}{4}d{\theta}=\frac{6561}  {4}(2{\pi})-\frac{6561}{4}(0)=\frac{6561\pi}{2}

    Now, multiply by -27: \frac{-177147\pi}{2}
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  5. #5
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    any help with this?

    help...

    int {[xy-4x^4y-ysqrt(x^2+y^2+1/2y^2sinx]dx

    + [4xy^4+8/3x^3y^2+sqrtx^2+y^2-ycosx]dy}

    where r is the path mapped counter-clockwise: along y=o,
    0<x<1; then along x^2+y^2=1, from (1,0)to (1/sqrt2,1/sqrt2);
    finally y=x from (1/sqrt2,1sqrt2) to (0,0)
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