Explanation Given: The region W is bounded in spherical coordinates by ϕ = ϕ 0 and the sphere ρ = R Formula Used: ( x ¯ , y ¯ , z ¯ ) = ( 1 V ∫ ∫ W ∫ x d V , 1 V ∫ ∫ W ∫ y d V , 1 V ∫ ∫ W ∫ z d V ) Calculation: The spherical coordinates are: x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ 0 ≤ ρ ≤ R , 0 ≤ θ ≤ 2 π , 0 ≤ ϕ ≤ ϕ 0 d V = ρ 2 sin ϕ d ρ d θ d ϕ Evaluating ∫ ∫ W ∫ d V , ∫ ∫ W ∫ d V = ∫ ϕ = 0 ϕ 0 ∫ θ = 0 2 π ∫ ρ = 0 R ρ 2 sin ϕ d ρ d ϕ d θ = [ 1 3 ρ 3 ] 0 R [ ∫ ϕ = 0 ϕ 0 sin ϕ d ϕ ] [ ∫ θ = 0 2 π d θ ] = [ 1 3 ( R 3 − 0 3 ) ] [ − cos ϕ ] ϕ = 0 ϕ 0 [ θ ] 0 2 π = [ 1 3 R 3 ] [ − cos ϕ 0 + cos 0 ] [ 2 π − 0 ] = 1 3 R 3 ( 1 − cos ϕ 0 ) ( 2 π ) = 2 π R 3 3 ( 1 − cos ϕ 0 ) Evaluating ∫ ∫ W ∫ x d V , ∫ ∫ W ∫ x d V = ∫ ϕ = 0 ϕ 0 ∫ θ = 0 2 π ∫ ρ = 0 R ( ρ sin ϕ cos θ ) ρ 2 sin ϕ d ρ d ϕ d θ = ∫ ϕ = 0 ϕ 0 ∫ θ = 0 2 π ∫ ρ = 0 R ρ 3 sin 2 ϕ cos θ d ρ d ϕ d θ = [ ∫ ρ = 0 R ρ 3 d ρ ] [ ∫ θ = 0 2 π cos θ d θ ] [ ∫ ϕ = 0 ϕ 0 sin 2 ϕ d ϕ ] = [ ρ 4 4 ] ρ = 0 R [ sin θ ] 0 2 π [ ∫ ϕ = 0 ϕ 0 sin 2 ϕ d ϕ ] = [ ρ 4 4 ] ρ = 0 R [ sin 2 π − sin 0 ] 0 2 π [ ∫ ϕ = 0 ϕ 0 sin 2 ϕ d ϕ ] = 0 Evaluating ∫ ∫ W ∫ y d V , ∫ ∫ W ∫ y d V = ∫ ϕ = 0 ϕ 0 ∫ θ = 0 2 π ∫ ρ = 0 R

4th Edition

ISBN: 9781319434540

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16, Problem 44CRE

To determine

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The centroid of the region $W$ .

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Chapter 16, Problem 43CRE

Chapter 16, Problem 45CRE

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Chapter 16 Solutions

ACHIEVE STANDALONE ACCESS F/ CALC 4E

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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The centroid of the region W .

Solution Summary: The author calculates the centroid of the region W. The spherical coordinates are x, y, and z.

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The centroid of the region $W$ .

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Chapter 16 Solutions

The centroid of the region W .

Solution Summary: The author calculates the centroid of the region W. The spherical coordinates are x, y, and z.

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To calculate: The centroid of the region W.

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Chapter 16 Solutions

To calculate:

The centroid of the region $W$ .