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

Describe in words the region of ${R}^{3}$ represented by the equation(s) or inequality.
${x}^{2}+{y}^{2}=4$
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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
That is,
$\left(x,y,z\right)|{x}^{2}+{y}^{2}=4,x\in R,y\in R,z\in R$
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.