I have a vector field $\vec{F} = (-2y, x)$ and the curve $\{(x, y) : x^2+y^2 \leq 1; y > |x|\}$. Now I want to calculate the work done by the vector field along the curve that goes positively.
So, all conditions for Green's theorem are met. I've done
$$\oint_C \vec{F}d\vec{c} = \int \int_R (\frac{\partial}{\partial x}(x) - \frac{\partial}{\partial y}(-2y))dA = 3\int_{\frac{\pi}{4}}^\frac{3\pi}{4}\int_0^1r^2rdrd\theta = \frac{3}{4}\int_{\frac{\pi}{4}}^\frac{3\pi}{4}d\theta = \frac{3\pi}{8}$$
But the solution gives $\frac{3\pi}{4}$. I don't really understand what I am doing wrong. I graphed the region and it seems good. I don't find any errors on my integral calculations as far as I can see. So I really don't see where I may be mistaken.
Thank you in advance!