# Thread: help with greens theorem

1. ## help with greens theorem

just trying figure this example out using greens theorem:

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

mapped counter-clockwise, is the circle

$
x^2 + y^2 = 81
$

enter as a multiple of pi

so using substitution,

$
\int {udx + vdy = \int_{} {\int_R {(\frac{{\delta y}}
{{\delta x}} - \frac{{\delta u}}
{{\delta x}}} } } )dxdy
$

so

$
u = 9y^3
$

$
v = - 9x^3
$

and

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

$
- 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...

2. Doesn't r vary from 0 to 9?.

$-27\int_{0}^{2\pi}\int_{0}^{9}r^{3}drd{\theta}$

3. 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

4. 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: $\frac{-177147\pi}{2}$

5. ## 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)