just trying figure this example out using greens theorem:

$\displaystyle

I = \oint_R {9y^3 dx - 9x^3 dy}

$

mapped counter-clockwise, is the circle

$\displaystyle

x^2 + y^2 = 81

$

enter as a multiple of pi

so using substitution,

$\displaystyle

\int {udx + vdy = \int_{} {\int_R {(\frac{{\delta y}}

{{\delta x}} - \frac{{\delta u}}

{{\delta x}}} } } )dxdy

$

so

$\displaystyle

u = 9y^3

$

$\displaystyle

v = - 9x^3

$

and

$\displaystyle

- 27\int {\int_R {x^2 + y^2 dxdy} }

$

$\displaystyle

- 27\int\limits_0^{2\pi } {\int\limits_0^1 {r^2 rdrd\theta } }

$

so i got this far how does this end bit work?? do i integrate? where do i go from here...