Many fewer tetrahedra are required. But as of now the 5 independent reals required to specify the tetrahedron plus three more to say where you are causes me to guess that the expression might be too expensive to evaluate, even if I could carry out the integrations in closed form. (dead end?)

A body with a ‘sort of curved surface’ is best approximated to some closeness by a polyhedron and any convex polyhedron with a boundary of n triangles can be expressed as the union of n tetrahedra. Non convex polyhedra too if we allow polyhedra of negative volume; the code need not even notice the sign.

Scheme code for simplex, simplex, about the code to form sphere from triangles.