I've had a go at calculating the contents of various toratopes using multiple integrals. I've worked out an interesting process for doing it in general, but it's not at a stage where I can do it algorithmically yet. The trick is to change into rotatopic coordinates, which reduces the problem to an integral of the Jacobian over a lower dimensional shape. In some cases this has to be done several times.

I'll try to post everything some time next week. In the meantime, here's a sneak preview. I'm writing in LaTeX notation because it's quicker for me, and I'm using tau = 2pi

We want the volume of (((II)(II))(III)I), with radii A, a_1, a_{11}, a_{12}, a_2.

Use the rotatopic coordinates for (II)(II)(III)I.

Then the volume is a product of surface contents of two circles and a sphere, multiplied by the integral of the Jacobian of the coordinate transform, over the toratope ((II)II) shifted by the vector (a_{11}, a_{12}, a_2, 0).

This requires another coordinate transform, this time using rotatopic coordinates (II)II. What I ended up with is:

V= 2 \tau^5 a_{11} a_{12} (2/15 A^5 + 2/3 A^3 a_2^2)