Explanation Given information: The region is bounded below by the xy -plane z = 0 and bounded above by the surface z = r . On both the sides, it is bounded by the cylinder with r = 4 . Calculation: The mass can be expressed as follows: M = ∭ D d V The first moment, second moment, and third moment can be expressed as, M x y = ∭ D z d V M x z = ∭ D y d V M y z = ∭ D x d V For cylindrical coordinates, x = r cos θ , y = r sin θ , z = z d V = d z r d r d θ and x 2 + y 2 = r 2 . The value of r = 4 . The limits in cylindrical coordinates of z is from 0 to r , limits of r is from 0 to 4, and limits of θ is from 0 to π 2 . Therefore, the mass M is M = 4 ∫ 0 π 2 ∫ 0 4 ∫ 0 r d z r d r d θ = 4 ∫ 0 π 2 ∫ 0 4 [ z ] 0 r r d r d θ = 4 ∫ 0 π 2 ∫ 0 4 [ r ] r d r d θ = 4 ∫ 0 π 2 ∫ 0 4 r 3 / 2 d r d θ M = 4 ∫ 0 π 2 [ 2 r 5 / 2 5 ] 0 4 d θ = 4 ∫ 0 π 2 [ 2 ( 4 ) 5 / 2 5 ] d θ = 256 5 ∫ 0 π 2 d θ = 256 5 [ θ ] 0 π 2 M = 256 5

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 14.7, Problem 93E

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Calculate the centroid of the bounded region.

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Chapter 14.7, Problem 92E

Chapter 14.7, Problem 94E

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Chapter 14 Solutions

University Calculus: Early Transcendentals (4th Edition)

Ch. 14.1 - In Exercises 1-14. evaluate the iterated...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - Prob. 10E

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