Explanation Given information: y = x + cos x y = 0 The range of limit is given by 0 ≤ x ≤ 2 π . Formula: M = ∫ a b y d x x ¯ = 1 M ∫ a b x y d x ∫ a b x a d x = [ x a + 1 a + 1 ] a b ∫ u ( x ) v ′ ( x ) d x = u ( x ) v ( x ) − ∫ v ( x ) u ′ ( x ) d x (1) cos 2 ( x ) = 1 + cos ( 2 x ) 2 ∫ a b cos a x d x = [ sin a x a ] a b ∫ a b sin x d x = [ − cos x ] a b Calculation: Calculate the mass M using the formula: M = ∫ a b y d x Substitute 0 for a , 2 π for b and x + cos x for y . M = ∫ 0 2 π ( x + cos x ) d x = ∫ 0 2 π x d x + ∫ 0 2 π cos x d x Apply the limits as below. M = ( x 2 2 ) 0 2 π + ( sin x ) 0 2 π = ( 2 π ) 2 2 + sin 2 π − sin 0 = 2 π 2 Calculate the x- coordinate x ¯ of centre of gravity about x axis using the below expression. x ¯ = 1 M ∫ a b x y d x Substitute 0 for a , 2 π for b, 2 π 2 for M and x + cos x for y . x ¯ = 1 2 π 2 ∫ 0 2 π x ( x + cos x ) d x = 1 2 π 2 ∫ 0 2 π ( x 2 + x cos x ) d x = 1 2 π 2 ( ∫ 0 2 π x 2 d x + ∫ 0 2 π x cos x d x ) x ¯ = 1 2 π 2 ( [ x 3 3 ] 0 2 π + ∫ 0 2 π x cos x d x ) (2) Consider the following integral term from equation (2). ∫ 0 2 π x cos x d x Let us take the term x as u ( x ) . u ( x ) = x Differentiate the term u with respect to x . u ′ ( x ) = d x Let us take the term cos x as v ′ ( x ) . Integrate the term with respect to x . ∫ v ′ ( x ) = ∫ cos x d x v ( x ) = sin x Substitute x for u ( x ) , d x for u ′ ( x ) , sin x for v ( x ) and cos x for v ′ ( x ) with the limits in equation (1). ∫ 0 2 π x cos x d x = [ x sin x ] 0 2 π − ∫ 0 2 π sin x d x = [ 2 π × sin 2 π − 0 ] − [ cos x ] 0 2 π = − [ 2 π × cos 2 π − cos 0 ] = 1 − 1 ∫ 0 2 π x cos x d x = 0 Substitute 0 for ∫ 0 2 π x cos x d x in equation (2). x ¯ = 1 2 π 2 [ x 3 3 ] 0 2 π = 1 2 π 2 [ 8 π 3 3 ] = [ 4 π 3 ] Calculate the y- coordinate y ¯ of centre of gravity about y axis: y ¯ = 1 M ∫ a b ∫ y d y d x Integrate the above equation with respect to y

Pearson eText University Calculus: Early Transcendentals -- Instant Access (Pearson+)

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ISBN: 9780136880912

Author: Joel Hass, Christopher Heil

Publisher: PEARSON+

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Chapter 8.2, Problem 75E

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