Calculating Volumes via Cross-Sections or General Slices

A 3-D printer builds a solid of unit height, inscribed in a unit cube, as follows. At height position $z\in [0,1]$, the cross section of the solid is a Y-shape (with three "legs" of width $x$) inscribed in a unit square, as shown above. (The "leg width", $x$, ranges from $0$ to $1/\sqrt2$ as $z$ ranges from $0$ to $1$.) Find the volume of the solid as a decimal rounded to the nearest hundredth.