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 + <

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ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.4, 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 16.4, Problem 45E

Chapter 16.4, Problem 47E

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Chapter 16 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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Prob. 29E Ch. 16.6 - Prob. 30E Ch. 16.6 - Prob. 31E Ch. 16.6 - Prob. 32E Ch. 16.6 - Prob. 33E Ch. 16.6 - Prob. 34E Ch. 16.6 - Prob. 35E Ch. 16.6 - Prob. 36E Ch. 16.6 - Prob. 37E Ch. 16.6 - Prob. 38E Ch. 16.6 - Prob. 39E Ch. 16.6 - Prob. 40E Ch. 16.6 - Prob. 41E Ch. 16.6 - Prob. 42E Ch. 16.6 - Prob. 43E Ch. 16.6 - Prob. 44E Ch. 16.6 - Prob. 45E Ch. 16.6 - Prob. 46E Ch. 16.6 - Prob. 47E Ch. 16.6 - Prob. 48E Ch. 16.6 - Prob. 49E Ch. 16.6 - Prob. 50E Ch. 16.6 - Prob. 51E Ch. 16.6 - Prob. 52E Ch. 16 - Prob. 1CRE Ch. 16 - Prob. 2CRE Ch. 16 - Prob. 3CRE Ch. 16 - Prob. 4CRE Ch. 16 - Prob. 5CRE Ch. 16 - Prob. 6CRE Ch. 16 - Prob. 7CRE Ch. 16 - Prob. 8CRE Ch. 16 - Prob. 9CRE Ch. 16 - Prob. 10CRE Ch. 16 - Prob. 11CRE Ch. 16 - Prob. 12CRE Ch. 16 - Prob. 13CRE Ch. 16 - Prob. 14CRE Ch. 16 - Prob. 15CRE Ch. 16 - Prob. 16CRE Ch. 16 - Prob. 17CRE Ch. 16 - Prob. 18CRE Ch. 16 - Prob. 19CRE Ch. 16 - Prob. 20CRE Ch. 16 - Prob. 21CRE Ch. 16 - Prob. 22CRE Ch. 16 - Prob. 23CRE Ch. 16 - Prob. 24CRE Ch. 16 - Prob. 25CRE Ch. 16 - Prob. 26CRE Ch. 16 - Prob. 27CRE Ch. 16 - Prob. 28CRE Ch. 16 - Prob. 29CRE Ch. 16 - Prob. 30CRE Ch. 16 - Prob. 31CRE Ch. 16 - Prob. 32CRE Ch. 16 - Prob. 33CRE Ch. 16 - Prob. 34CRE Ch. 16 - Prob. 35CRE Ch. 16 - Prob. 36CRE Ch. 16 - Prob. 37CRE Ch. 16 - Prob. 38CRE Ch. 16 - Prob. 39CRE Ch. 16 - Prob. 40CRE Ch. 16 - Prob. 41CRE Ch. 16 - Prob. 42CRE Ch. 16 - Prob. 43CRE Ch. 16 - Prob. 44CRE Ch. 16 - Prob. 45CRE Ch. 16 - Prob. 46CRE Ch. 16 - Prob. 47CRE Ch. 16 - Prob. 48CRE Ch. 16 - Prob. 49CRE Ch. 16 - Prob. 50CRE Ch. 16 - Prob. 51CRE Ch. 16 - Prob. 52CRE Ch. 16 - Prob. 53CRE Ch. 16 - Prob. 54CRE Ch. 16 - Prob. 55CRE Ch. 16 - Prob. 56CRE Ch. 16 - Prob. 57CRE Ch. 16 - Prob. 58CRE Ch. 16 - Prob. 59CRE Ch. 16 - Prob. 60CRE Ch. 16 - Prob. 61CRE

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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 16 Solutions