Explanation Given: Hemisphere x 2 + y 2 + z 2 ≤ R 2 , z ≥ 0 . Formula Used: Centroid is given by: ( x ¯ , y ¯ , z ¯ ) = ( 1 V ∭ W x d V , 1 V ∭ W y d V , 1 V ∭ W z d V ) ... ( 1 ) Volume of the hemisphere is: V = 2 3 π R 3 Calculation: Assume, density δ ( x , y ) = 1 . The region in spherical form 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 ≤ R , 0 ≤ θ ≤ 2 π , 0 ≤ ϕ ≤ π 2 . First evaluate ∭ W x d V and we get: ∭ W x d V = ∫ ϕ = 0 π 2 ∫ θ = 0 2 π ∫ ρ = 0 R ( ρ sin ϕ cos θ ) ρ 2 sin ϕ d ρ d θ d ϕ = ∫ ϕ = 0 π 2 ∫ θ = 0 2 π ∫ ρ = 0 R ( ρ 3 sin 2 ϕ cos θ ) d ρ d θ d ϕ = [ ρ 4 4 ] 0 R [ ∫ 0 π 2 sin 2 ϕ d ϕ ] [ ∫ 0 2 π cos θ d θ ] = R 4 4 [ ∫ 0 π 2 1 − cos 2 ϕ 2 d ϕ ] [ sin θ ] 0 2 π = R 4 4 [ 1 2 ( ϕ − 1 2 sin 2 ϕ ) ] 0 π 2 [ 0 ] = R 4 4 ( π 4 ) [ 0 ] = 0 Now, evaluate ∭ W y d V and we get: ∭ W y d V = ∫ ϕ = 0 π 2 ∫ θ = 0 2 π ∫ ρ = 0 R ( ρ sin ϕ sin θ ) ρ 2 sin ϕ d ρ d θ d ϕ = ∫ ϕ = 0 π 2 ∫ θ = 0 2 π ∫

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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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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Chapter 16 Solutions

CALCULUS W/SAPLING ACCESS >IC<

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 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 $δ (x, y) = 1$ .