# Describe in words the region of R^{3} represented by the equation(s) or inequality. x^{2} + y^{2} = 4

Question
Alternate coordinate systems
Describe in words the region of $$R^{3}$$ represented by the equation(s) or inequality.
$$x^{2} + y^{2} = 4$$

2020-12-30

Concept:
The equation $$x^{2} + y^{2} = r^{2}$$ represents a circle whose centre lies on z - axis
Given:
$$x^{2} + y^{2} =4$$
3) Calculation:
The given equation is
$$x^{2} + y^{2} = 4$$
The equation $$x^{2} + y^{2} =4$$
represents the set of all points in $$R^{3}\ lying\ on\ circle\ x^{2} + y^{2} = 4.$$
That is,
$${(x, y, z) | x^{2} + y^{2} = 4, x \in R, y \in R, z \in R}$$
Here is no restriction on z-coordinate, so a point in the region must lie on a circle with radius 2 and centre on z-axis but it could be any horizontal plane $$z = k$$ (parallel to xy - plane)
Therefore, the region consists of all points on the circle $$x^{2} + y^{2} = 4, z = k$$
That is, a circular cylinder with radius 2 whose axis is the z - axis
Therefore, the given equations represents the region in $$R^{3}$$ consisting of all possible citcles of radius 2 and centre on z -axis that 1s a circular cylinder with radius 2 whose axis is the z - axis.
Conclusion:
The given equations represents the region in $$R^{3}$$ consisting of all possible circles of radius 2 and centre on z - axis that is a circular cylinder with radius 2 whose axis is the z - axis.

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