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**HallsofIvy** I find it very difficult to read your math "expressions". You say you want to "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".

Okay, in the first quadrant $\displaystyle \theta$ goes from 0 to $\displaystyle \pi/2$. Converting the first circle to polar coordinates again is easy- $\displaystyle r^2= 4$ means that r= 2 (r is never negative).

The other circle is not much harder: $\displaystyle x^2+ y^2 $ is again $\displaystyle r^2$ and $\displaystyle 2x= 2r cos(\theta)$ so the equation is $\displaystyle r^2= 2rcos(\theta)$ or ( if r is not 0, $\displaystyle r= 2cos(\theta)$. So the double integral is $\displaystyle \int_{\theta= 0}^{\pi/2}\int_{r= 2}^{2cos(\theta)} (r cos(\theta)) r dr d\theta$