I'm not being able to visualize this region.
I don't have or know any kind of software that can graph this:
$$R = \{(x,y,z) \in \mathbb R^3 \mid (-1 \le y \le 1) \wedge (y^2 \le x \le 1) \wedge (0 \le z \le x) \}.$$
I did a few drawings but I'm not convinced.
Could anyone graph this for me and tell me the used software?
On
This weirdo is your region, complementing bob.sacamento's answer.
It was made in Wolfram Mathematica with the command:
RegionPlot3D[-1 < y < 1 && y^2 < x < 1 && 0 < z < x, {x, 0, 1}, {y, -2, 2}, {z, 0, 1}]
Well, it's hard to visualize because it is -- apparently -- very arbitrary. But I'll try to help:
Your first condition is $y$ between -1 and 1. Imagine two planes, each parallel to the $x-z$ plane, one at $y=-1$ and one at $y=1$.
For your second condition, concentrate on the $x-y$ plane and imagine plotting $x=1$ and $y^2=x$ (or $y=x^{1/2}$). $x$ lies in the area between these two plots. Imagine both plots extended vertically in the $z$ direction. That's your second condition.
Third, imagine a plane that contains the $y$ axis, and everywhere on the plane, $z=x$. Also imagine the $xy$ plane (where $z=0$). The volume between those two planes is your third condition.
The intersection of these volumes is your range.
Hope that helps.