Its quite a big question sorry... but I really am just looking for the starting point for three cylinders (not two sorry) on a 2D plane. Hopefully after that I can find the common volume for either unequal radii or at different angles like the photo below. Thank you for your help!
A shot in the dark of what I have so far.. (not too sure if it is of use) Cylinder equations $$x^2 + y^2 = r_1^2$$ $$x^2 + z^2 = r_2^2$$ $$y^2 + z^2 = r_3^2$$ where $r_1 \neq r_2 \neq r_3$
The common volume for infinite cylinders with equal radii using triple integration with circular co-ordinates is: $$V_c = 8\cdot(2-\sqrt 2)\cdot r^3$$
(following this great video)