I am unsure what you mean by "x dA". I will proceed assuming there is no density function (nothing actually inside the integral besides dxdy). In this case,
So the first circle is, in polar coordinates:
The second circle is:
It helps to graph the two curves. You'll find that in the first quadrant you will be integrating from the smaller circle to the bigger circle:
Here, theta from 0 to Pi/2 indicates the first quadrant, and r from to 2 indicates you are going from the small circle to the big circle.
This is easily checkable since we are dealing with perfect circles. If you look at the plot you see that you are subtracting half of a circle with radius 1 from 1/4 of a circle of radius 2. Thus:
Now, if you meant that you have a density function f(x) = x to integrate over, it changes things a bit: