   Chapter 15.8, Problem 15E

Chapter
Section
Textbook Problem

A solid lies above the cone z = x 2 + y 2 and below the sphere x2 + y2 + z2 = z. Write a description of the solid in terms of inequalities involving spherical coordinates.

To determine

To write: The description of given solid by the inequalities in spherical coordinates.

Explanation

Given:

The solid lies above the cone z=x2+y2 and below the sphere x2+y2+z2=z .

Formula used:

The rectangular coordinates (x,y,z) corresponding to the spherical coordinates (ρ,θ,ϕ) is,

x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕ

Calculation:

Obtain the value of x2+y2 using the formula mentioned above.

x2+y2=(ρsinϕcosθ)2+(ρsinϕsinθ)2=ρ2sin2ϕcos2θ+ρ2sin2ϕsin2θ=ρ2sin2ϕ(cos2θ+sin2θ)=

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