Solution. The geometry of the domain of integration is illustrated below. The green parabolic cylinder is intersected by the plane x + z = 1
As for the inequalities that define the region: First, fix y, where
.
Then x starts from the parabola (at
)
and runs to the line x = 1 ( blue line). That is,
.
For fixed x , z runs from 0 to 1- x (red line). Thus, the volume integral
is
