   Chapter 16.6, Problem 25E

Chapter
Section
Textbook Problem

Find a parametric representation for the surface.25. The part of the cylinder x2 + y2 + z2 = 36 that lies between the planes z = 0 and z = 3 To determine

To find: The parametric representation for the part of the sphere x2+y2+z2=36 that lies between the planes z=0 and z=33 .

Explanation

Given data:

The equation of part of the sphere is given as follows.

x2+y2+z2=36x2+y2+z2=62

From the equation of the sphere, the radius of the sphere is 6.

The required part of the sphere lies between the planes z=0 and z=33 .

Formula used:

Write the expression for parametric equations of the sphere with radius r .

x=rsinϕcosθ,y=rsinϕsinθ,z=rcosϕ (1)

Here, the limits of θ and ϕ are written as follows.

ϕlowerϕϕupperθlowerθθupper

Calculation of parametric equations of the sphere:

Substitute 6 for r in equation (1),

x=6sinϕcosθ,y=6sinϕsinθ,z=6cosϕ

Calculation of ϕ limits:

As the required part of the sphere lies between the planes z=0 and z=33 , equate z-parameter to the given values and find the lower and upper limits of ϕ .

Case-i: If z=0

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