My question is the same as this question except that I am looking for the intersection volume of two caps instead of the area.
Given a sphere with radius $R$ that has two spherical caps with base radii $a_1$ and $a_2$, find the intersection volume of the two spherical caps. Let $\theta$ be the angle that separates the two caps, defined as the angle between the two lines $L_1$ and $L_2$ that go through the centre of the respective caps and the centre of the sphere.
At first, I tried looking at the formula for a spherical wedge but realised that the 'tip' of the wedge is not always at the centre. My second idea was to try and define an integral in spherical coordinates but I quickly ran into issues because I cannot figure out how to adjust the radius properly as the radius range will depend on the angle. I can find the angle between the two line intersections (from the circles of the caps) on the surface of the sphere but these are not necessarily correct, because going towards the centre of the sphere from these intersection lines again creates a wedge (that is much larger than the intersection volume).
The geometry is shown here.
Thank you in advance.