Calculate the double integral: $\iint_D\frac{1}{\sqrt{x^2+y^2}}dxdy$ with $D=\{(x,y)\in\ R^2:4y \le x^2+y^2; 4 \le x^2+y^2 \le 16; y \ge 0; x \ge 0\}$

Question

I am trying to solve the following double integral using polar coordinates:

$$\iint_D\frac{1}{\sqrt{x^2+y^2}}dxdy$$ $$D=\{(x,y)\in\ R^2:4y \le x^2+y^2; 4 \le x^2+y^2 \le 16; y \ge 0; x \ge 0\}$$

But upon graphing the region represented by $D$, I am unable to determine the intervals as I am not sure how to work with the intersection of the two inner circles.

Any suggestions?

Answer 1

Find the point $P$ then the angle and separate the integral into the two separate regions.