Homework question, so just hints please
Sketch the region $$ R=\{ (x,y) \in \mathbb{R} : y \geq x, 1 \leq x^2 + y^2 \leq 2 \} $$ and, by changing to polar coordinates, compute $$ \iint \limits_R \frac{xy}{x^2 + y^2} \mathrm{d}x \, \mathrm{d}y $$
So I have the sketch, as from WolframAlpha:
and so, in polar coordinates, the region is described by $$ R=\{ (r,\theta) \in \mathbb{R} : 1 \leq r \leq 2, \frac{\pi}{4} \leq \theta \leq \frac{5 \pi}{4} \} $$
and thus we have the integral $$ \begin{align*} \iint \limits_R \frac{xy}{x^2 + y^2} \mathrm{d}x \, \mathrm{d}y &= \int_1^2 \int_\frac{\pi}{4}^\frac{5 \pi}{4} \! \frac{r\cos\theta \, r\sin\theta}{(r\cos\theta)^2 + (r\sin\theta)^2} \mathrm{d}\theta \, \mathrm{d}r\\ &= \int_1^2 \int_\frac{\pi}{4}^\frac{5 \pi}{4} \! \sin\theta \, \cos\theta \, \mathrm{d}\theta\, \mathrm{d}r\\ \end{align*} $$ but then clearly this is wrong as it evaluates to $0$. However, I don't see where I have made a mistake.
What you have done seems to be correct except that the area element $dxdy$ becomes $r dr d\theta$ in polar coordinates, and the maximum value of r is $\sqrt{2}$. However, this will still give 0 but what's wrong with that?