Calculating using Direct and Green's Theorem

[bugatti79]

Homework Statement

Calculate the folowing directly and with greens theorem

Homework Equations

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

C= x^2+y^2=4

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 can't spot the error!

 

[LCKurtz]

$\displaystyle \int \int_R (G_x -F_y)dA= \int_0^{2 \pi}\int_0^2 2 r dr d \theta = 2 \pi$...? I can't spot the error!

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

 

[bugatti79]

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

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

Thanks LCKurtz