Rearrange the implicit equation and observe symmetry and intersections to simplify the task.
vol = int(int(abs(z1)/2,x,0,1),y,0,1)
h1 = fsurf(z0,[0 1 0 1]);
h2 = fsurf(z1,[0 1 0 1]);
I'm not going to bother pruning the surfaces or plotting the vertical faces. This should suffice to make the volume understandable. There are also plenty of other examples of plotting piecewise surfaces using standard tools like surf().