Question

In Exercises \(15-20,\) find an equation in rectangular coordinates for the equation given in cylindrical coordinates, and sketch its graph. $$ r^{2}+z^{2}=4 $$

Short answer

The equivalent equation in rectangular coordinates is \( x^2 + y^2 + z^2 = 4 \), which represents a sphere with the centre at the origin and radius 2.

Step-by-step solution

Understanding Cylindrical and Rectangular Coordinate systems

The rectangular coordinate system is based on x, y and z coordinates while the cylindrical coordinate system uses r, θ and z where r is the radius in circular base (which is formed by x and y in rectangular), θ is the angle formed by x-axis and line from origin to the point in base and z is height or depth from this base.

Converting the given cylindrical coordinates to rectangular coordinates

The given equation is \( r^2 + z^2 = 4 \). In the cylindrical coordinate system, the radius r in the circular base is given by \( r = \sqrt{x^2 + y^2} \). By substituting this into the given equation, we get \( (\sqrt{x^2 + y^2})^2 + z^2 = 4 \). Simplifying this gives \( x^2 + y^2 + z^2 = 4 \).

Identifying the graph of the equation

This equation represents the equation of a sphere in three dimensions. The sphere has a center at the origin (0,0,0) and has a radius of 2, since 4 is the square of the radius of the sphere. Therefore, the graph would comprise of a sphere centered at origin and having radius equal to 2.