I remember the South African Police being asked about integration in the late 1970's, in those dark racist days there.
The Police Chief said there was nothing to worry about, since: "our boys are up all night integrating suspects"...
Must get a snack - all this talking has given me quite an apartheid.....
I was doing something similar by trying to add two segments with similar chord length but differing radii. It all went t*ts up when it dawned that I had a second variable (the angle) in the equations - at which point I realised I had forgotten about integrals.
I was thinking of doing it by a different way. The area of sector + the areas of the 2 segments. Rather curious if we can still remember the A level maths done decades ago. But on second thought, I would rather make some lunch instead.
With no one volunteering a solution I decided to see if the answer was online anywhere - and it is. Here is a comprehensive solution on wikipedia along with several further problems and solutions on the same theme:
https://en.wikipedia.org/wiki/Goat_problem
I didn't know it was quite THAT complicated!