Explanation Given: Region is a sphere of radius 2 from which a central cylinder of radius 1 has been removed. Formula used: Using spherical co-ordinates the volume is V = ∭ ρ 2 sin ϕ d ρ d ϕ d θ Calculation: Equation of sphere of radius 2 is ρ = 2 and equation of cylinder is of radius 1 with center at origin is ρ sin ϕ = 1 . The intersection points of above solids are given by 1 sin ϕ = 2 ⇒ sin ϕ = 1 2 ⇒ ϕ = π 6 , 5 π 6 Now using above formula required volume is V = ∫ 0 2 π ∫ π 6 5 π 6 ∫ 1 sin ϕ 2 ρ 2 sin ϕ d ρ d ϕ d θ = ∫ 0 2 π ∫ π 6 5 π 6 ( 1 3 ρ 3 ) 1 sin ϕ 2 sin ϕ d ϕ d θ = 1 3 ∫ 0 2 π ∫ π 6 5 π 6

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.4, Problem 44E

To determine

The volume of the sphere of radius 2 from which a central cylinder of radius 1 has been removed, using spherical coordinates.

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Chapter 16.4, Problem 43E

Chapter 16.4, Problem 45E

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Chapter 16 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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The volume of the sphere of radius 2 from which a central cylinder of radius 1 has been removed, using spherical coordinates.

Solution Summary: The author calculates the volume of the given region using spherical coordinates. The intersection points of above solids are given by 1 sin = 2

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The volume of the sphere of radius 2 from which a central cylinder of radius 1 has been removed, using spherical coordinates.

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Chapter 16 Solutions