The task is as follows:
Find double integral (in polar coordinates) over region $D$ of $f(x,y) = (x^2 + y)$ by $dxdy$.
The region $D$ is bounded by a circle of radius $3$ and a circle of radius $2$, with upper angle $a1$ being $\frac{\pi}{4}$ and lower angle $a2$ being $\frac{\pi}{6}$
The region is as the sketch below:
I need help on setting up the double integral.
My work so far:
(1) Let $x = r \cos \theta$ and $y = r \sin \theta$
(2) The double integral in polar coordinates should be something in nature of $$ \int_{\theta_1} ^ {\theta_2} \int_{r_1}^{r_2} f(r\cos\theta,r\sin\theta) r dr d{\theta}$$
(3) Base on the sketch of the region, $r$ should go from $2$ to $3$,
But I don't see the lower and upper bounds for $\theta$. Should it be from $0$ to $a1 + a2$?
Or should I consider two integrals instead? One with $\theta$ from $0$ to $a1$ and another integral with $\theta$ from $0$ to $a2$?
I just want to make sure that my original thoughts are right (or not).
Would someone please help me on this?
Thank you.