Question # To determine: Find the sets of points in space whose coordinates satisfy the given combinations of equation and inequalities: a) displaystyle{y}ge{x}^{2},{z}ge{0}, b) displaystyle{x}le{y}^{2},{0}le{z}le{2}.

Alternate coordinate systems
ANSWERED To determine:
Find the sets of points in space whose coordinates satisfy the given combinations of equation and inequalities:
a) $$\displaystyle{y}\ge{x}^{2},{z}\ge{0},$$
b) $$\displaystyle{x}\le{y}^{2},{0}\le{z}\le{2}.$$ 2020-11-06

a)
Generally, the plane is determined by the coordinated axis is the xy-plane, whose standard equation by $$\displaystyle{z}={0}$$.
From equation $$\displaystyle{y}\ge{x}^{2},{z}\ge{0},{y}={x}^{2}$$ represent the parabola curve in the xy-plane and all point above the region $$\displaystyle{z}={0}.$$
Figure 1 shows for $$\displaystyle{y}\ge{x}^{2},{z}\ge{0}$$ in the Cartesian coordinate system. Thuse, the sets of points in space whose coordinates satisfy the given combinations of equations and inequalities for $$\displaystyle{y}\ge{x}^{2},{z}\ge{0}$$ is the region inside the parabola $$\displaystyle{y}\ge{x}^{2}$$ in the xy-plane.
b)
From equation $$\displaystyle{x}\le{y}^{2},{0}\le{z}\le{2},{x}={y}^{2}$$ represent the parabola curve in the xy-plane in the region on or to the left of the parabola and also have all point above the region from $$\displaystyle{z}={0}\to{z}={2}.$$
Figure 2 shows for $$\displaystyle{x}\le{y}^{2},{0}\le{z}\le{2}$$ in the Cartesian coordinate system. Thus, the sets of points in space whose coordinates satisfy the given combinations of equations and inequalities for $$\displaystyle{x}\le{y}^{2},{0}\le{x}\le{2}$$ is the region on or to the left the parabola $$\displaystyle{x}\le{y}^{2}$$ in the xy-plane and all points from $$\displaystyle{z}={0}\to{z}={2}.$$