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.)

To  answer this question, we should make use of spherical coordinates.

Solution is:

S = (  4×cosθ ×sinΦ , 4 ×sinθ× sinΦ, 4 × cosΦ)

0 ≤ θ ≤ 2×π     ;  π/2  ≤ Ф ≤ (3/2)×π

In Analitic Geometry we have different way of determine, and identify  the position of objects, we have rectangular coordinates, cylindrical coordinates and spherical coordinates. The use of each of these system depends on de geometry of the problem.

In this particular case and according to the problem statement we should use spherical coordinates

x = ρ×cosθ ×sinΦ           y = ρ ×sinθ× sinΦ         z = ρ× cosΦ

In our particular case

ρ = 4   then   x = 4×cosθ ×sinΦ     y = 4 ×sinθ× sinΦ   z = 4 × cosΦ

0 ≤ θ ≤ 2×π     ;   π/2  ≤ Ф ≤ ( 3/2)×π

So the solution in terms of θ and/or  Φ

S = (  4×cosθ ×sinΦ , 4 ×sinθ× sinΦ, 4 × cosΦ)

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The standard choice would be the usual representation in spherical coordinates. Let

We get the part of the sphere between the planes with and .

step-by-step explanation: