Evaluate the integral given that ∫ ∫ x dA, where D is the region in the first quadrant that lies between the circles x^2 + y^2 = 4 and x^2 + y^2 = 2x (D is under the double integral)

so I decided it would be easier to use polar coordinates for this problem. My limits for theta are 0 to Pi/2 and I used pythag to find the first limit of r which is 2, My professor gave us a hint and told us for x^2 + y^2 = 2x , r = 2cos(theta). Is there a way to actually find the other limit or is this just some kind of rule?

I also set up the double integral

Pi/2 2

∫ . . ∫ rcos(theta) *r drd(theta)

0 . . 2cos(theta)

I integrated the r but I think I integrated incorrectly, I got this

Pi/2 . . . . . . . . . . . . . . . . . . 2

∫ (r^2/2)cos(theta) *(r^2/2)| d(theta)

0 . . . . . . . . . . . . . . . . . . . . 2cos(theta)

I'm not sure how I should go about integrating the first r in front of the cosine.