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.
(<a href=BCC.html>dead end?</a>)
<p>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.
<hr><a href=cnfInt.scm>Scheme code for simplex</a>,
<a href=Simplex.html>simplex</a>,
<a href=ccode.html#sph>about the code to form sphere</a> from triangles.