Inside your definition of cortes there is this:
Column[{TraditionalForm[Text[Style[ineq, Blue, Bold, 14]]],
Show[{g}, Boxed -> bx, Axes -> ax, ViewPoint -> vp,
ImageSize -> 500, AxesStyle -> axst, BoxStyle -> bxst]}, Center,
Dividers -> {None, {1 -> None, 2 -> GrayLevel[0.7]}}]
Inside it can you see ineq
? Replace ineq
with
Row[{ineq, ", volume = ",
RegionMeasure[
ImplicitRegion[
eqn[x, y,
z], {{x, xmin, xmax}, {y, ymin, ymax}, {z, zmin, zmax}}]]}]
It works on my system: the volume is shown in the Manipulate
window.