Explanation Given: Region bounded by the xy-plane, the cylinder x 2 + y 2 = R 2 , and the plane x R + z H = 1 ,where R > 0 and H > 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 cylindrical form is: x = r cos θ , y = r sin θ , z = z , d V = r d z d r d θ and 0 ≤ r ≤ R , 0 ≤ θ ≤ 2 π , 0 ≤ z ≤ H ( 1 − r R cos θ ) . First evaluate V = ∭ W d V and we get: ∭ W d V = ∫ θ = 0 2 π ∫ r = 0 R ∫ z = 0 H ( 1 − r R cos θ ) r d z d r d θ = ∫ θ = 0 2 π ∫ r = 0 R r [ z ] 0 H ( 1 − r R cos θ ) d r d θ = ∫ θ = 0 2 π ∫ r = 0 R r [ H ( 1 − r R cos θ ) ] d r d θ = H ∫ 0 2 π [ r 2 2 − 1 R cos θ ( r 3 3 ) ] 0 R d θ = H ∫ 0 2 π [ R 2 2 − 1 R cos θ ( R 3 3 ) ] d θ = H R 2 2 ( 2 π ) − 0 = π R 2 H Now, evaluate ∭ W x d V and we get: ∭ W x d V = ∫ θ = 0 2 π ∫ r = 0 R ∫ z = 0 H ( 1 − r R cos θ ) r 2 cos θ d z d r d θ = ∫ θ = 0 2 π ∫ r = 0 R r 2 cos θ [ z ] 0 H ( 1 − r R cos θ ) d r d θ = ∫ θ = 0 2 π ∫ r = 0 R r 2 cos θ [ H ( 1 − r R cos θ ) ] d r d θ = H ∫ 0 2 π ∫ 0 R cos θ ( r 2 − r 3 R cos θ ) d r d θ = H ∫ 0 2 π [ r 3 3 cos θ − 1 R cos 2 θ ( r 4 4 ) ] 0 R d θ = H ∫ 0 2 π [ R 3 3 cos θ − 1 R cos 2 θ ( R 4 4 ) ] d θ = H R 3 3 [ ∫ 0 2 π cos θ d θ ] − H R 3 4 [ ∫ 0 2 π cos 2 θ d θ ] = 0 − π H R 3 4 Now, evaluate ∭ W y d V and we get: ∭ W y d V =

4th Edition

ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 16.5, Problem 20E

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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 19E

Chapter 16.5, Problem 21E

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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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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 $δ (x, y) = 1$ .