Explanation Given information: f ( x , y , z ) = 1 x 2 + y 2 + z 2 , region 5 ≤ x 2 + y 2 + z 2 ≤ 25 Formula used: For a continuous function on the region θ 1 ≤ θ ≤ θ 2 , ϕ 1 ≤ ϕ ≤ ϕ 2 , ρ 1 ( θ , ϕ ) ≤ ρ ≤ ρ 2 ( θ , ϕ ) , the triple integral ∭ W f ( x , y , z ) d V = ∫ θ 1 θ 2 ∫ ϕ 1 ( θ ) ϕ 2 ( θ ) ∫ ρ 1 ( r , θ ) ρ 2 ( r , θ ) f ( ρ sin ϕ cos θ , ρ sin ϕ sin θ , ρ cos ϕ ) ρ 2 sin ϕ d ρ d ϕ d θ Calculation: First, write f ( x , y , z ) = 1 x 2 + y 2 + z 2 in spherical coordinates Thus, f ( x , y , z ) = 1 x 2 + y 2 + z 2 = 1 ( ρ sin ϕ cos θ ) 2 + ( ρ sin ϕ sin θ ) 2 + ( ρ cos ϕ ) 2 = 1 ρ 2 ( sin 2 ϕ ( cos 2 θ + sin 2 θ ) + cos 2 ϕ ) = 1 ρ 2 ( sin 2 ϕ ( 1 ) + cos 2 ϕ ) = 1 ρ 2 ( sin 2 ϕ + cos 2 ϕ ) = 1 ρ 2 Now, express the region in spherical coordinates, 5 ≤ x 2 + y 2

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

4th Edition

ISBN: 9781319312886

Author: Rogawski, Jon, Adams, Colin, FRANZOSA, Robert

Publisher: W. H. Freeman

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Chapter 15.4, Problem 46E

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Calculate the triple integral of f(x,y,z)=1x2+y2+z2, over 5≤x2+y2+z2≤25 using spherical coordinates.

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Chapter 15.4, Problem 45E

Chapter 15.4, Problem 47E

Chapter 15 Solutions

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

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Calculate the triple integral of f ( x , y , z ) = 1 x 2 + y 2 + z 2 , over 5 ≤ x 2 + y 2 + z 2 ≤ 25 using spherical coordinates.

Solution Summary: The author explains how to calculate the triple integral of f(x,y,z)=1x2+y

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Calculate the triple integral of f(x,y,z)=1x2+y2+z2, over 5≤x2+y2+z2≤25 using spherical coordinates.

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Chapter 15 Solutions