Explanation Consider the density ( δ ( x , y , z ) ) is 1 and 2 units on a side. The solid is bounded by the planes x = ± 1 , z = ± 1 , y = 3 , and y = 5 . Calculation: Calculate the mass of the solid ( M ) as shown below. M = ∭ D δ d V Substitute 1 for δ , dzdydx for dV , and apply the limits. M = ∫ − 1 1 ∫ 3 5 ∫ − 1 1 d z d y d x = ∫ − 1 1 ∫ 3 5 ( z ) − 1 1 d y d x = ∫ − 1 1 ( 1 − ( − 1 ) ) ( y ) 3 5 d y d x = 2 ∫ − 1 1 ( 5 − 3 ) d x = 4 ( x ) − 1 1 = 4 ( 1 − ( − 1 ) ) = 8 Calculate the first moment ( M y z ) as shown below. M y z = ∭ D δ x d V Substitute 1 for δ , dzdydx for dV , and apply the limits. M y z = ∫ − 1 1 ∫ 3 5 ∫ − 1 1 x d z d y d x = ∫ − 1 1 ∫ 3 5 ( z ) − 1 1 x d y d x = ∫ − 1 1 ( 1 − ( − 1 ) ) x ( y ) 3 5 d y d x = 2 ∫ − 1 1 x ( 5 − 3 ) d x = 4 ( x 2 2 ) − 1 1 = 2 ( 1 − ( − 1 ) 2 ) = 0 Calculate the first moment ( M x z ) as shown below. M x z = ∭ D δ y d V Substitute 1 for δ , dzdydx for dV , and apply the limits. M x z = ∫ − 1 1 ∫ 3 5 ∫ − 1 1 y d z d y d x = ∫ − 1 1 ∫ 3 5 ( z ) − 1 1 y d y d x = ∫ − 1 1 ( 1 − ( − 1 ) ) ( y 2 2 ) 3 5 d y d x = ∫ − 1 1 ( 5 2 − 3 2 ) d x = 16 ( x ) − 1 1 = 16 ( 1 − ( − 1 ) ) = 32 Calculate the first moment ( M x y ) as shown below. M x y = ∭ D δ z d V Substitute 1 for δ , dzdydx for dV , and apply the limits. M x y = ∫ − 1 1 ∫ 3 5 ∫ − 1 1 z d z d y d x = ∫ − 1 1 ∫ 3 5 ( z 2 2 ) − 1 1 d y d x = 1 2 ∫ − 1 1 ( 1 2 − ( − 1 ) 2 ) d y d x = 0 Calculate the center of mass ( x ¯ ) as shown below. x ¯ = M y z M Substitute 0 for M y z and 8 for M . x ¯ = 0 Calculate the center of mass ( y ¯ ) as shown below. y ¯ = M x z M Substitute 32 for M x z and 8 for M . y ¯ = 32 8 = 4 Calculate the center of mass ( z ¯ ) as shown below. z ¯ = M x y M Substitute 0 for M x y and 8 for M . z ¯ = 0 Hence, the center of mass is ( x ¯ , y ¯ , z ¯ ) = ( 0 , 4 , 0 ) _ . Calculate the moment of inertia about x -axis ( I x ) as shown below. I x = ∭ D ( y 2 + z 2 ) δ d V Substitute 1 for δ , dzdydx for dV , and apply the limits

University Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780321999580

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 14.6, Problem 26E

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Calculate the center of mass and the moments of inertia about the coordinate axes.

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Chapter 14.6, Problem 25E

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University Calculus: Early Transcendentals (3rd Edition)

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