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: enter image description here

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.


1 Answer 1


the angular range will be as given from $-\frac{\pi}{6}$ to $\frac{\pi}{4}$ assuming $a1$ is the angle above the $x-$axis not including the angle $a2$, otherwise the upper value is $\frac{\pi}{4} - \frac{\pi}{6} = \frac{\pi}{12}$


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