Line Integrals

1. Jan 20, 2012

bugatti79

1. The problem statement, all variables and given/known data
Calculate the folowing directly and with greens theorem

2. Relevant equations

$\int (x-y) dx + (x+y) dy$

C= x^2+y^2=4

3. The attempt at a solution

Directly

$x= r cos \theta, y=r sin \theta, r^2=4, dx = -r sin \theta d \theta, dy= r cos \theta d \theta$

Substituting I get

$\displaystyle \int_0^{2 \pi} (-r^2 sin \theta cos \theta +r^2 sin^2 \theta) d \theta+(r^2 cos^2 \theta +r^2 sin \theta cos \theta) d \theta$

$=4 \int_0^{2 \pi} d \theta= 8 \pi$

Greens theorem

$\displaystyle \int \int_R (G_x -F_y)dA= \int_0^{2 \pi}\int_0^2 2 r dr d \theta = 2 \pi$.....? I cant spot the error!

2. Jan 20, 2012

LCKurtz

I can't spot the error either because you didn't show your [incorrect] work to get $2\pi$.

3. Jan 21, 2012

bugatti79

I have spotted it this morning. Just used wrong limits in calculation although shown correctly above. Late night concentration I guess.

Thanks LCKurtz