Application of Green's theorem in a plane

Question

Apply Green's theorem in a plane to evaluate $\displaystyle\int(2x^2-y^2)dx + (x^2+y^2)dy$, where $C$ is the curve enclosed by the semi-circle $x^2+y^2=1$ and the $x$-axis.

I've done this already $$ \int Pdx + Qdy = \iint (dQ/dx - dP/dy) \text{ } dxdy $$

$P=(2x^2-y^2)$

$Q=(x^2+y^2)$

$\displaystyle\frac{dQ}{dx}=2x$

$\displaystyle\frac{dP}{dy}=-2y$

$$ \iint (2x+2y)dxdy $$

So how do I get the range for $x$ and $y$ to integrate with respect to?

Answer 1

You've done fine so far. The boundary of the region is the semicircle described, so the region you need for the double is simply the interior of that semicircle. In polar coordinates $0\leq r\leq1,\ 0\leq\theta\leq\pi.$