When we are given
and we are given a region in the xy domain bounded by a circle of radius and by a circle of some other unknown radius (we need to figure this out). Further, we should note that the second circle is actually shifted on the x-axis, so to properly conclude what radius we need in our integral, we should draw this all out.
What we can do is complete the square,
Now we can see our radius clearly. Our radius is and the circle is centered at
If we add our radius plus our shift along the x-axis, we can find the distance this circle travels as to see if it lies inside or outside the radius of the circle centered at the origin.
It so happens that so we cant tell much from this, other then that our circles in the xy domain overlap along the x-axis at the very most right end. But if you select a value of x that is minimally smaller then and sub it into both equations, you will find that the radius of the second circle is now less then . Thus, we are going from the inside of our shifted circle to the edge of our regular circle
To most apply model this we can go back to the original equation and switch to polars,
Of course leading to,