Based on what I see in your link, your ROC volume can be subdividing into individual five-faced polyhedrons each of whose base is a triangle in the x-y plane, whose three sides are trapezoids orthogonal to this plane and whose top is another (slanted) triangle. Your volume is the sum of the volumes of these separate polyhedrons.
Call the cartesian coordinates of the vertices of a top triangle (x1,y1,z1), (x2,y2,z2), and (x3,y3,z3) where the ordering from points 1 to 2 to 3 is counterclockwise with respect to the x-y plane. Then the volume of the polyhedron underneath this top triangle is:
V = 1/6*det([x1,x2,x3;y1,y2,y3;1,1,1])*(z1+z2+z3)
(The area of its triangular base is 1/2*det([x1,x2,x3;y1,y2,y3;1,1,1]) and the average height is 1/3*(z1+z2+z3).) Note that if the ordering of the three points is clockwise, this expression will be the negative of the actual volume.
Thus, your job is now to determine the coordinates of all those points in the surface.
(Corrected)
