Explanation The density of thin plate ( δ ( x , y ) ) is 1 + x 20 . The plate bounded by the lines x = 0 , x = 20 , y = − 1 , and y = 1 . Calculation: Calculate the mass of the plate ( M ) as shown below. M = ∬ R δ d A Substitute 1 + x 20 for δ , dydx for dA , and apply the limits. M = ∫ 0 20 ∫ − 1 1 ( 1 + x 20 ) d y d x = ∫ 0 20 ( 1 + x 20 ) ( y ) − 1 1 d x = ∫ 0 20 ( 1 + x 20 ) ( 1 − ( − 1 ) ) d x = 2 ∫ 0 20 ( 1 + x 20 ) d x = 2 ( x + 1 20 x 2 2 ) 0 20 = 2 ( 20 + 20 2 40 − 0 ) = 60 Calculate the first moment ( M y ) as shown below. M y = ∬ R x δ d A Substitute 1 + x 20 for δ , dydx for dA , and apply the limits. M y = ∫ 0 20 ∫ − 1 1 ( 1 + x 20 ) x d y d x = ∫ 0 20 ( x + x 2 20 ) ( y ) − 1 1 d x = ∫ 0 20 ( x + x 2 20 ) ( 1 − ( − 1 ) ) d x = 2 ∫ 0 20 ( x + x 2 20 )

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 14.6, Problem 18E

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Calculate the center of mass and mass moment of inertia about the x-axis of a thin plate.

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Chapter 14.6, Problem 17E

Chapter 14.6, Problem 19E

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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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Calculate the center of mass and mass moment of inertia about the x- axis of a thin plate.

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Chapter 14 Solutions