Green's Theorem for a circle

[manenbu]

Homework Statement

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

Homework Equations

Green's Theorem.

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)?

 

[mighty2000]

Your solution is correct. The minus sign in front of $\frac{\pi}{2}$ is correct because Green's theorem states that the line integral is equal to the negative of the double integral. As for the transformation to polar coordinates, it is up to personal preference. Both ways are valid, but using polar coordinates simplifies the integration.