Using Green's Theorem to Solve a Circle Line Integral

[Liferider]

Homework Statement

Use greens theorem to solve the closed curve line integral:
$\oint$(ydx-xdy)

The curve is a circle with its center at origin with a radius of 1.

Homework Equations

x^2 + y^2 = 1

The Attempt at a Solution

Greens theorem states that:
Given F=[P,Q]=[y, -x]=yi-xj

$\oint$F*dr=$\oint$Pdx+Qdy=$\int$$\int$($\frac{dQ}{dx}$-$\frac{dP}{dy}$)dA

From the circle equation i find:
x=$\sqrt{1-y^2}$
y=$\sqrt{1-x^2}$

Which means that:
$\frac{dQ}{dx}$=0 and $\frac{dP}{dy}$=0

Obviously, I am doing something wrong... but which rules am I breaking??

I did find the answer the "normal" way, without greens, which was -2$\pi$.

I think a lot of my difficulties originates from lack of knowledge about different coordinate systems and the conversion between these.
One could write:
x=cos t and y=sin t, but do I have to? (btw, i used that for solving the normal way).

 

[Liferider]

Oh wait... I think I got the vector field F mixed up with the circle that I am integrating over.
I shouldn't insert for x and y before I have applied greens theorem... I think.

 

[Quinzio]

It's $Q = -x, P= y$, so $\frac{dQ}{dx} = -1$, etc...

 

[Liferider]

Solved, used polar coordinates on the integral after greens theorem.