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