Trouble finding the limits of integration for polar coordinates

Question

Use polar coordinates to evaluate $\iint_D x \, dA $, where D is the region inside the circle $x^2 +(y-1)^2 = 1$ but outside the circle $x^2 +y^2 = 1$ as shown below.

Hi all, i'm stuck on finding the limits of integration for this particular question. The question states to use polar coordinates so what i've got was the following

$\pi/6 \le \theta \le {5\pi}/6$ and $1 \le r \le ??$

The upper limit of $r$ that i've gotten was something like $\sqrt{2+2\sqrt{1-r^2cos^2\theta}}$ which doesn't seem right.

Could anyone give me a hint on how to get the upper limit of r? Thanks in advance

Answer 1

Hint Substitute the usual rectangular-to-polar transformation rules $x = r \cos \theta$, $y = r \sin \theta$, in the equation $$x^2 + (y - 1)^2 = 1$$ defining the upper circle.

Answer 2

The range of $r$ is $1\le r\le 2\sin\theta$