How do I define $\theta$?

Question

I need to calculate $\iint f(x,y) \,dx \,dy$ with the given region $D$ where $$f(x,y) = xy$$ and $$D = \{(x,y)\in \mathbb{R^2} \space | \space 0 \leq y \leq x, x^2+y^2 \leq 4 \}$$

I have used cylindrical coordinates trying to calculate this. So $f(x,y) = r \cos(\theta) r \sin(\theta)$ and because $x^2+y^2 \leq 2^2$ is a circle, I know that $ 0 \leq r \leq 2$. Now, how do I know the values for $\theta$? I can't get anything out of $0 \leq y \leq x$.

Answer 1

I found an answer to my question thanks to Winther. Below you can see a picture of the region $D$. As you can see when we sketch $y=x$, we divide the circle and we know that $$0 \leq \theta \leq \frac{\pi}{4}.$$

So $$f(x,y) = \int_0^{\frac{\pi}{4}}\int_0^2 r\cos(\theta)r \sin(\theta) r \,dr \,d\theta = 1.$$ Thanks Winther!