Draw a scetch of the original set. You should see its a circle with a parabola on top.

Further your boundary on this set is exactly there where equality holds in the original set so its

{z = 0 and x^2 + y^2 <= 1} and { z = 1 - x^2 - y^2 }

Both of them are finite 2d manifolds so are null sets in 3d.