   Chapter 14.7, Problem 31E

Chapter
Section
Textbook Problem

VolumeIn Exercises 31–34, use spherical coordinates to find the volume of the solid.Solid inside x 2 + y 2 + z 2 = 9 , outside z = x 2 + y 2 , and above the x y -plane

To determine

To calculate: Volume using spherical coordinates of the solid inside sphere x2+y2+z2=9 outside z=x2+y2 and above x-y plane.

Explanation

Given:

Equation of solid inside sphere x2+y2+z2=9 outside z=x2+y2 and above x-y plane

Formula Used:

Rectangular conversion components for spherical coordinates

x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕx2+y2+z2=ρ2

Volume of solids in polar form

Vf(x,y,z)dV=Vf(ρsinϕcosθ,ρsinϕsinθ,ρcosϕ)ρ2sinϕdρdϕdθ

Calculation:

From given equation of sphere,

x2+y2+z2=9ρ=3 …… (1)

And

z=x2+y2z2=x2+y2 …… (2)

Use (2)

z2+z2=92z2=9z=32ρcosϕ=32

From (1), Bounds on ϕ are

When ρ=3

3cosϕ=32cosϕ=12

When 0ρ3

π4ϕπ2

Bounds on ρ are

0ρ3

Bounds on θ are

0θ2π

Volume in Spherical coordinates

V=Vρ2sinϕdρdϕdθ=02ππ4

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