# Green's Theorem for a circle

1. Sep 19, 2009

### manenbu

1. The problem statement, all variables and given/known data

Use Green's Theorem to calculate second type line integral:
$$I = \oint_{\Gamma} x^2 y dx - xy^2dy$$
where $\Gamma[/tex] is the edge of domain [itex] D = \left\{(x,y) | x^2 + y^2 \leq 1, y \geq 0 \right\}$

2. Relevant equations

Green's Theorem.

3. The attempt at a solution

Ok, so according to Green's Therom this integral becomes:
$$-\iint_{D} x^2 + y^2 dA$$
And since my domain is the upper part of a radius 1 circle centered at the origin, we can make it like this:
$$-\int_{0}^{\pi} \int_{0}^{1}(\cos^2{\theta} + \sin^2{\theta}) r dr d\theta = -\int_{0}^{\pi} \int_{0}^{1} r dr d\theta = -\frac{\pi}{2}$$

But for some reason I got a feeling that something went wrong here, maybe because of the minus sign in front of $\frac{\pi}{2}$?
Was I correct in making the transformation to polar coordinates after constructing the double integral, or should it had been better before (even though the differentiation becomes much more complicated then)?