Double integral with polar coordinates, circle within a circle

Question

I am trying to integrate the following:

$$\iint_G(x+y)^2\,dx\,dy$$

With $$ G = \{(x,y) : y\geq x, 4\leq x^2+y^2 \leq 9\}.$$

I know how it works, but the answers show that you need to use $$\frac54\pi \text{ and } \frac14\pi $$

to calculate the double integral.

Can somebody explains how they come up with these $\pi$ numbers?

Thanks :)

Answer 1

For a point on the line $y=x$ the polar coordinates are $ (r,\frac {\pi} 4)$ if $x >0$ and $ (r,\frac {5\pi} 4)$ if $x <0$, $r$ being $\sqrt {x^{2}+y^{2}}$. For points with $y \geq x$, $\theta$ starts at $\frac {\pi} 4$ and ends at $\frac {5\pi} 4$.