Help with line integral using Green's theorem

Question

$$ \int_\Gamma \frac{x \; dy-y \; dx}{x^2+y^2} $$ where $\Gamma$ is a circle: $x^2+y^2-2x-2y+1=0$

By "completing the square" I see that the circle has a radius of $1$ and is moved one point to the right and one up. Partial derivatives of $M(x,y)$ and $N(x,y)$ are: $$ \frac{\partial M}{\partial y}=\frac{y^2-x^2}{(x^2+y^2)^2} $$ $$ \frac{\partial N}{\partial x}=\frac{y^2-x^2}{(x^2+y^2)^2} $$ So, I get $$ \int\int \left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) \; dA = \int\int dA $$ Am I correct up to this point? How do I continue? What should be the bounds for r if I convert it into polar coordinates?

Answer 1

Your working seems right.

Did you notice that you got the same terms, and now you are trying to integrate their difference?

Answer 2

The integral $\displaystyle\int\int dA$ is just the area of the circle, so if you're allowed to cite other ways of finding that than doing the integral in polar coordinates, then you've got it.

However, if you subtract something from itself, you get $0$, so if you've got $\displaystyle\int\int 0\;dA$, then the integral is $0$.