Okay I am following but wouldn't the bounds for r be r=0 and r=sqrt(3/4)? When you find z=1/2 you also need to find the distance from the center at z=1/2 to the outer boundary where the surfaces of the spheres intersect because this is less than 1. Am I correct on this?
If two spheres of radius 1 intersect each other so the surface of each sphere passes through the other’s center how could you find the exact volume of the intersection? It doesn't matter what kind of method, it could be double or triple integration, geometry, polar, cylindrical or spherical...