Polar coordinate double integral

Question

I have to integrate the following integral:

$$ \iint \limits_A sin({x_1}^2 + {x_2}^2) dx_1dx_2 $$

over the set: $A=\{x \in \mathbb{R}^2: 1 \leq {x_1}^2 + {x_2}^2 \leq 9,x_1 \geq -x_2\}$

I understand geometrically what I have to do (i.e. subtract one half of a circle with of radius 1 from half of a circle of radius 3) but how do I use polar coordinates to calculate this interal?

Answer 1

Hint: You can use that $1\le r\le 3 \text{ and } -\frac{\pi}{4}\le\theta\le\frac{3\pi}{4}$ to set up the integral.