Question

Find a parametric representation for the surface.

The part of the sphere

x² + y² + z² = 16

that lies between the planes

z = −2 and z = 2.

(Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of θ and/or ϕ.)

Answer 1

To find a parametric representation for the surface of the sphere that lies between the planes z = -2 and z = 2, we can use spherical coordinates. Let's denote the spherical coordinates as (r, φ, θ), where r is the radial coordinate, φ is the polar angle (relative to the vertical z-axis), and θ is the azimuthal angle (relative to the x-axis).

Step-by-step explanation:

To find a parametric representation for the surface of the sphere that lies between the planes z = -2 and z = 2, we can use spherical coordinates. Let's denote the spherical coordinates as (r, φ, θ), where r is the radial coordinate, φ is the polar angle (relative to the vertical z-axis), and θ is the azimuthal angle (relative to the x-axis).

The parametric representation for the surface is x = rsin(φ)cos(θ), y = rsin(φ)sin(θ), and z = rcos(φ), where 0 ≤ φ ≤ π and 0 ≤ θ ≤ 2π.

Since the sphere has the equation x² + y² + z² = 16, we can express x, y, and z in terms of φ and θ as follows:

x = rsin(φ)cos(θ)

y = rsin(φ)sin(θ)

z = rcos(φ)

Now we just need to find the appropriate range for φ and θ that satisfies the condition z = -2 and z = 2. Since the sphere lies between these two planes, we have -2 ≤ z ≤ 2. Therefore, the value of φ can range from 0 to π, and the value of θ can range from 0 to 2π.