   Chapter 16.6, Problem 23E

Chapter
Section
Textbook Problem

Find a parametric representation for the surface.23. The part of the sphere x2 + y2 + z2 = 4 that lies above the cone z = x 2 + y 2

To determine

To find: The parametric representation for part of the sphere x2+y2+z2=4 that lies above the cone z=x2+y2 .

Explanation

Given data:

The equation of the sphere is x2+y2+z2=4 and it lies above the cone z=x2+y2 .

Consider x and y as parameters and parameterize the equation of sphere as follows.

Rewrite the equation of sphere x2+y2+z2=4 as follows.

x2+y2+z2=4z2=4x2y2z=±4x2y2

As the required part of the sphere lies above the cone z=x2+y2 , the parameter of z must be a positive value.

z=+4x2y2

Therefore, the parameters of part of the sphere x2+y2+z2=4 are written as follows.

x=xy=yz=4x2y2

The limit of the parameters is determined as follows.

Consider the equation of sphere as follows.

x2+y2+z2=4 (1)

Write the equation of cone that intersects the sphere as follows

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