How do I find the centroid of a lens?

Question

A lens is a convex-convex area bounded by two circular arcs. I have a large circle centered at the origin and a smaller circle centered at (d,0). The center of the smaller circle is between (0,0) and (rad_large, 0), top left image. I need to find the centroid of the area shared by both circles. I have thought about finding the centroid of the lune and using that with the centroid of the small circle to find the centroid of the area, but I don't know the centroid of the lune, as stated in this question.

Answer 1

The centroid is the center of mass of the entire figure. Obviously, the $($horizontal$)$ line uniting the centers of the two circles splits the lens $($and lune$)$ into two equal halves, so it clearly contains the centroid. Now we must find another line, perpendicular on the first, which also cuts the lens into two parts of equal area. Let's say that the larger circle is centered at the origin, and has radius R, while the smaller one is centered at $(a,0)$, and has radius r. We must then find u so that $$\int_{a-r}^u\sqrt{r^2-(x-a)^2}~dx~=~\int_u^X\sqrt{r^2-(x-a)^2}~dx~+~\int_X^R\sqrt{R^2-x^2}~dx,$$ where $X=\dfrac a2+\dfrac{R^2-r^2}{2a}$ is the horizontal coordinate of the two points of intersection of the two circles.