# To write: The inequalites to describe the region of a solid cylinder that lies on or below the plane displaystyle{z}={8} and on or above the disk in the xy - plane with the center at origin and radius 2.

Question
Alternate coordinate systems
To write: The inequalites to describe the region of a solid cylinder that lies on or below the plane $$\displaystyle{z}={8}$$ and on or above the disk in the xy - plane with the center at origin and radius 2.

2020-12-25
$$\displaystyle\mathbb{R}^{3}$$ is the three dimensional coordinate system which contains x, y, and z - coordinates.
The equation $$\displaystyle{z}={8}\in\mathbb{R}^{3}$$ represents the set $$\displaystyle{\left\lbrace{\left({x},{y},{z}\right)}{|}{z}={8}\right\rbrace}$$, which is the set of all points in $$\displaystyle\mathbb{R}^{3}$$ whose z - coordinate is 8 and x, y - coordinates are any values.
The inequality to describe the region in which all points lie on or below the plane $$\displaystyle{z}={8}\ {i}{s}\ {0}\le{z}\le{8}$$.
The equation to describe the region of cylinder with center at the origin and radius of 2 units on the xy-plane is $$\displaystyle{x}^{2}+{y}^{2}={4}$$. But, it is required to describe the region in which all points lie on or above the disk in the xy - plane.
The inequality to describe the region of solid cylinder that lies on or above the disk in the xy - plane wiwth center at origin and radius of 2 units $$\displaystyle{x}^{2}+{y}^{2}\le{4}.$$
Thus, the inequality to describe the region of solid cylinder that lies on or below the plane $$\displaystyle{z}={8}$$ and on or above the disk in the xy - plane with center at the origin and radius $$\displaystyle{2}\ {i}{s}\ {x}^{2}+{y}^{2}\le{4},{0}\le{z}\le{8}.$$

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