Ask question

# 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

Question
Alternate coordinate systems asked 2020-12-29
Describe in words the region of $$R^{3}$$ represented by the equation(s) or inequality.
$$x^{2} + y^{2} = 4$$

## Answers (1) 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.

### Relevant Questions asked 2021-03-02

Descibe in words the region of $$\mathbb{R^{3}}$$ represented by the equation or inequality.
$$y = -2$$ asked 2021-03-25
Describe in words the region $$\displaystyle{\mathbb{{{R}}}}^{{3}}$$ represented by the equation:
$$\displaystyle{X}{Y}{Z}={0}$$ asked 2021-02-22
Describe in words the region of $$R^{3}$$ represented by the equation(s) or inequality.
$$x = 5$$ asked 2020-11-23
Describe in words the region of $$RR^{3}$$ represented by the equation or inequality.
$$z \geq -1$$ asked 2021-01-30

A line L through the origin in $$\displaystyle\mathbb{R}^{{3}}$$ can be represented by parametric equations of the form x = at, y = bt, and z = ct. Use these equations to show that L is a subspase of $$RR^3$$  by showing that if $$v_1=(x_1,y_1,z_1)\ and\ v_2=(x_2,y_2,z_2)$$  are points on L and k is any real number, then $$kv_1\ and\ v_1+v_2$$  are also points on L. asked 2021-05-16
Solve the equation $$\log(x+3)+\log x =1$$, inequality, or system of equations. asked 2021-05-13
Find a polar equation for the curve represented by the given Cartesian equation.
$$x^{2}+y^{2}=100$$ asked 2021-01-31

(10%) In $$R^2$$, there are two sets of coordinate systems, represented by two distinct bases: $$(x_1, y_1)$$ and $$(x_2, y_2)$$. If the equations of the same ellipse represented by the two distinct bases are described as follows, respectively: $$2(x_1)^2 -4(x_1)(y_1) + 5(y_1)^2 - 36 = 0$$ and $$(x_2)^2 + 6(y_2)^2 - 36 = 0.$$ Find the transformation matrix between these two coordinate systems: $$(x_1, y_1)$$ and $$(x_2, y_2)$$. asked 2020-12-24
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. asked 2020-12-17

A surface is represented by the following multivariable function,
$$\displaystyle{f{{\left({x},{y}\right)}}}=\frac{{1}}{{3}}{x}^{{3}}+{y}^{{2}}-{2}{x}{y}-{6}{x}-{3}{y}+{4}$$
a) Calculate $$\displaystyle{f}_{{x x}},{f}_{{{y}{x}}},{f}_{{{x}{y}}}{\quad\text{and}\quad}{f}_{{{y}{y}}}$$
b) Calculate coordinates of stationary points.
c) Classify all stationary points.

...