You need an area function in terms of x only.
Let the equation of the circle be:
and let x=k be the coordinate from the center to the end on the right.
The y coordinate in the first quadrant is .
Split the cross into two parts:
The horizontal band including the shared square with the vertical band.
This horizontal band has length 2k and height 2y, so its area is:
The upper half of the vertical band has area given by 2y*height where height is k - y.
Combine A and 2B (note: it's 2B because you need the lower half of the vertical band as well), then do calculus with it.
Does this help?