Explanation Given information: The density function δ = 1 . The function g ( x ) = x 2 ( x + 1 ) . The function f ( x ) = 2 . The value of x = 0 . Calculation: Show the functions as follows. g ( x ) = x 2 ( x + 1 ) (1) f ( x ) = 2 (2) Calculate the intersection points as shown below. g ( x ) = f ( x ) x 2 ( x + 1 ) = 2 x 3 + x 2 − 2 = 0 ( x + 1 ) 2 ( x − 2 ) = 0 x = − 1 or x = 1 Sketch the thin plate bounded by the functions g ( x ) = x 2 ( x + 1 ) and f ( x ) = 2 as shown in Figure 1. Refer to Figure 1. Calculate the center of mass ( x ˜ , y ˜ ) as shown below. ( x ˜ , y ˜ ) = ( x , 1 2 ( f ( x ) + g ( x ) ) ) Substitute x 2 ( x + 1 ) for g ( x ) and 2 for f ( x ) . ( x ˜ , y ˜ ) = ( x , 1 2 ( x 2 ( x + 1 ) + 2 ) ) = ( x , 1 2 ( x 3 + x 2 + 2 ) ) Calculate the length of the plate as shown below. L = f ( x ) − g ( x ) Substitute x 2 ( x + 1 ) for g ( x ) and 2 for f ( x ) . L = 2 − x 2 ( x + 1 ) = 2 − x 3 − x 2 The width is dx . Calculate the area dA as shown below. d A = L × width Substitute 2 − x 3 − x 2 for L and dx for width. d A = ( 2 − x 3 − x 2 ) d x Calculate the mass of the plate as shown below. d m = δ d A Substitute 1 for δ and ( 2 − x 3 − x 2 ) d x for dA . d m = ( 2 − x 3 − x 2 ) d x Calculate the mass of the plate as shown below. ∫ d m = ∫ 0 1 ( 2 − x 3 − x 2 ) d x M = ( 2 x − x 4 4 − x 3 3 ) 0 1 = ( 2 × 1 − 1 4 4 − 1 3 3 ) − 0 = 2 − 1 4 − 1 3 = 24 − 3 − 4 12 = 17 12 Calculate δ 2 [ f 2 ( x ) − g 2 ( x ) ] as shown below. Substitute x 2 ( x + 1 ) for g ( x ) and 2 for f ( x ) . δ 2 [ f 2 ( x ) − g 2 ( x ) ] = 1 2 ( 2 2 − ( x 3 + x 2 ) 2 ) d x = 1 2 ( 4 − x 6 − x 4 − 2 x 5 ) d x Calculate the moment of the plate about x axis as shown below

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 6.6, Problem 38E

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Calculate the centroid of the thin plate.

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Chapter 6.6, Problem 37E

Chapter 6.6, Problem 39E

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University Calculus: Early Transcendentals (4th Edition)

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