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.