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
asked 2020-11-05
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) \(\displaystyle{x}\le{y}^{2},{0}\le{z}\le{2}.\)

Answers (1)

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.
image
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.
image
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 \(\displaystyle{z}={0}\to{z}={2}.\)

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