Explanation Given: Show the integral function as follows: ∫ a b ( c x + d ) f ( x ) d x = ( c x ¯ + d ) ∫ a b f ( x ) d x The integral function of left hand side (LHS) is ∫ a b ( c x + d ) f ( x ) d x . Calculation: The ( x ¯ ) coordinate of centroid is expressed as follows: x ¯ = 1 A ∫ a b x f ( x ) d x (1) Rearrange Equation (1). x ¯ A = ∫ a b x f ( x ) d x The area of the function as follows: A = ∫ a b f ( x ) d x The integral function as follows: ∫ a b ( c x + d ) f ( x ) d x = ∫ a b [ c x f ( x ) + d f ( x ) ] d x = ∫ a b c x f ( x ) d x + ∫ a b d f ( x ) d x = c ∫ a b x f ( x ) d x + d ∫ a b f ( x ) d x (2) Substitute x ¯ A for ∫ a b x f ( x ) d x in Equation (2)

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ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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Chapter 8.3, Problem 43E

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To prove: The integral function as ∫ab(cx+d)f(x)dx=(cx¯+d)∫abf(x)dx

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Chapter 8.3, Problem 42E

Chapter 8.3, Problem 44E

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To prove: The integral function as ∫ a b ( c x + d ) f ( x ) d x = ( c x ¯ + d ) ∫ a b f ( x ) d x

Solution Summary: The author explains how the integral function of left hand side (LHS) is proved.

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To prove: The integral function as ∫ab(cx+d)f(x)dx=(cx¯+d)∫abf(x)dx

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Chapter 8 Solutions