Explanation Given: The “ice cream cone” region W bounded, in spherical coordinates, by the cone ϕ = π 3 and the sphere ρ = 2 . Formula Used: Centroid is given by: ( x C M , y C M , z C M ) = ( M y M , M x M , M z M ) ... ( 1 ) M = ∬ D δ ( x , y , z ) d A M y z = ∬ D x δ ( x , y , z ) d A , M x z = ∬ D y δ ( x , y , z ) d A and M x y = ∬ D z δ ( x , y , z ) d A Calculation: Assume, density δ ( x , y ) = 1 . The spherical coordinate is: x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ , d V = ρ 2 sin ϕ d ρ d θ d ϕ and 0 ≤ ρ ≤ 2 , 0 ≤ θ ≤ 2 π , 0 ≤ ϕ ≤ π 3 . First evaluate M = ∭ W ρ d V and we get: M = ∭ W ρ d V = ∫ 0 2 π ∫ 0 π 3 ∫ ρ = 0 2 ρ 2 sin ϕ d ρ d ϕ d θ = ∫ 0 2 π ∫ 0 π 3 [ ρ 3 3 ] 0 2 sin ϕ d ϕ d θ = [ 8 3 ] ∫ 0 2 π [ − cos ϕ ] 0 π 3 d θ = 8 3 ∫ 0 2 π [ − 1 2 + 1 ] d θ = 4 3 [ θ ] 0 2 π = 4 3 [ 2 π − 0 ] = 8 π 3 Now, evaluatex-coordinate M y z = ∭ W x ρ d V and we get: M y z = ∭ W x ρ d V = ∫ 0 2 π ∫ 0 π 3 ∫ ρ = 0 2 ( ρ sin ϕ cos θ ) ρ 2 sin ϕ d ρ d ϕ d θ = ∫ 0 2 π ∫ 0 π 3 [ ρ 4 4 ] 0 2 sin 2 ϕ cos θ d ϕ d θ = ∫ 0 2 π ∫ 0 π 3 4 sin 2 ϕ cos θ d ϕ d θ = 4 [ ∫ 0 2 π cos θ d θ ∫ 0 π 3 sin 2 ϕ d ϕ ] = 4 [ sin θ ] 0 2 π

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.5, Problem 19E

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The centroid of the given solid region assuming a density of $δ (x, y) = 1$ .

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Chapter 16.5, Problem 18E

Chapter 16.5, Problem 20E

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The centroid of the given solid region assuming a density of δ ( x , y ) = 1 .

Solution Summary: The author calculates the centroid of the given solid region assuming a density of (x, y ) = 1.

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The centroid of the given solid region assuming a density of $δ (x, y) = 1$ .

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Chapter 16 Solutions

The centroid of the given solid region assuming a density of δ ( x , y ) = 1 .

Solution Summary: The author calculates the centroid of the given solid region assuming a density of (x, y ) = 1.

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To Calculate: The centroid of the given solid region assuming a density of δ(x,y)=1.

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Chapter 16 Solutions

To Calculate:

The centroid of the given solid region assuming a density of $δ (x, y) = 1$ .