Explanation Definition: Suppose f ( x ) is continuous on [ a , b ] and Δ x width of interval, then the function ∫ a b f ( x ) d x = lim n → ∞ ∑ i = 1 n f ( x i * ) Δ x . The sum ∑ i = 1 n f ( x i * ) Δ x is called the Riemann sum. Property used: 1. If f ( x ) ≥ g ( x ) for a ≤ x ≤ b , then ∫ a b f ( x ) d x ≥ ∫ a b g ( x ) d x . 2. Let n be a positive integer and c a real number, then ∑ i = 1 n c = n . Calculation: The integral function is ∫ 2 6 f ( x ) d x . Here, f ( x ) ≥ 5 for all x ∈ [ 2 , 6 ] . That is, g ( x ) = ∫ 2 6 5 d x . Compute the inequality ∫ 2 6 f ( x ) d x ≥ g ( x ) by substituting g ( x ) = ∫ 2 6 5 d x in the property (1) mentioned above. ∫ 2 6 f ( x ) d x ≥ ∫ 2 6 5 d x (1) The function is defined over the interval [ 2 , 6 ] . The interval [ 2 , 6 ] is divided into n subintervals of length Δ x = b − a n = 4 n . That is, Δ x = 4 n . Here, the Right Riemann sum ∑ i = 1 n f ( x i * ) Δ x is used to solve ∫ 2 6 5 d x , where x i * is the right endpoint of [ x k − 1 , x k ]

To choose: The appropriate option for which value is less than or equal to ∫ 2 6 f ( x ) d x if f ( x ) ≥ 5 for all x ∈ [ 2 , 6 ] .

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

8th Edition

ISBN: 9781305279148

Author: Stewart, James, St. Andre, Richard

Publisher: Cengage Learning

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Chapter 5.2, Problem 5PT

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To choose: The appropriate option for which value is less than or equal to ∫26f(x) dx if f(x)≥5 for all x∈[2,6].

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Show that if ax2+bx+c=0 for all x, then a=b=c=0.

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To choose: The appropriate option for which value is less than or equal to ∫ 2 6 f ( x ) d x if f ( x ) ≥ 5 for all x ∈ [ 2 , 6 ] .

Solution Summary: The author explains that the Riemann sum is the appropriate option for the value which is less than or equal to displaystyle.

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To choose: The appropriate option for which value is less than or equal to ∫26f(x) dx if f(x)≥5 for all x∈[2,6].

Chapter 5 Solutions