I'm wondering how to compute the area of a duocylinder. Specifically, I'd like to be able to do it using calculus. I know how to calculate areas of a surface in 3D, and volumes of a 4D object, but an area in 4D is difficult, because we don't have a cross product.

I think all we have to do is find the area of a parallelogram, determined by two 4D vectors. Does anyone know how to do this?