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

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

Nannie Mack 2020-12-29 Answered

Describe in words the region of

R^{3}

represented by the equation(s) or inequality.

x^{2} + y^{2} = 4

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Expert Answer

tafzijdeq

Answered 2020-12-30 Author has 92 answers

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 $R^{3} l y i n g o n c i r c l e x^{2} + y^{2} = 4.$
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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