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.

0

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.

...