There is a formula for this. What we have is two spherical caps.

The formula for the volume of a spherical cap is

h=the height of the cap. In this case it is 3/2.

Since we have two of them, we multiply by 2 and get

Also, there is a general formula for the volume of two intersecting spheres.

Where d is the distance between the radii, in this case 3.

You could derive this using calcarooney if you wish, but when we have a formula, why bother?.