Explanation Given information: The density function δ = 1 . The function g ( x ) = 0 . The function f ( x ) = 2 + sin x . The value of x = 0 and x = 2 π . Calculation: Show the functions as follows. g ( x ) = 0 (1) f ( x ) = 2 + sin x (2) Sketch the thin plate bounded by the functions g ( x ) = 0 and f ( x ) = 2 + sin x 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 0 for g ( x ) and 2 + sin x for f ( x ) . ( x ˜ , y ˜ ) = ( x , 1 2 ( 2 + sin x + 0 ) ) = ( x , 1 2 ( 2 + sin x ) ) Calculate the length of the plate as shown below. L = f ( x ) − g ( x ) Substitute 0 for g ( x ) and 2 + sin x for f ( x ) . L = 2 + sin x − 0 = 2 + sin x The width is dx . Calculate the area dA as shown below. d A = L × width Substitute 2 + sin x for L and dx for width. d A = ( 2 + sin x ) d x Calculate the mass of the plate as shown below. d m = δ d A Substitute 1 for δ and ( 2 + sin x ) d x for dA . d m = ( 2 + sin x ) d x Calculate the mass of the plate as shown below. ∫ d m = ∫ 0 2 π ( 2 + sin x ) d x M = ( 2 x − cos x ) 0 2 π = ( 2 × 2 π − cos ( 2 π ) − ( 0 − cos 0 ) ) = 4 π − 1 − ( 0 − 1 ) = 4 π Calculate the δ 2 [ f 2 ( x ) − g 2 ( x ) ] as shown below. Substitute 0 for g ( x ) and 2 + sin x for f ( x ) . δ 2 [ f 2 ( x ) − g 2 ( x ) ] = 1 2 ( ( 2 + sin x ) 2 − 0 ) d x = 1 2 ( 2 + sin x ) 2 d x = 1 2 ( 4 + sin 2 x + 4 sin x ) d x Calculate ∫ a b δ 2 [ f 2 ( x ) − g 2 ( x ) ] d x as shown below

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ISBN: 9780134175706

Author: Unknown

Publisher: PEARSON

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Chapter 6.6, Problem 32E

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Calculate the centroid of the thin plate.

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Chapter 6.6, Problem 31E

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