Explanation Given information: ∫ 1 5 x x 2 + 1 2 d x , n = 8 . Formula used: L n = b − a n ⋅ f a + f x 1 + f x 2 + ... + f x n − 2 + f x n − 1 R n = b − a n ⋅ f x 1 + f x 2 + ... + f x n − 2 + f x n − 1 + f b Calculation: Let f x = x x 2 + 1 2 , then the width of each subinterval is, b − a n = 5 − 1 8 = 0.5 Since n = 8 , these are eight rectangles. Calculate left Riemann sum L 8 The areas of eight rectangles can be calculated in the following table: Subinterval Width Height of rectangle at left end point x Area of rectangle =Width × Height 1 , 1.5 0.5 f 1 = 1 1 2 + 1 2 = 4 0.5 × 4 = 2 1.5 , 2 0.5 f 1.5 = 1.5 1.5 2 + 1 2 = 15.84375 0.5 × 15.84375 2 , 2.5 0.5 f 2 = 2 2 2 + 1 2 = 50 0.5 × 50 2.5 , 3 0.5 f 2.5 = 2.5 2.5 2 + 1 2 = 131.40625 0.5 × 131.40625 3 , 3.5 0.5 f 3 = 3 3 2 + 1 2 = 300 0.5 × 300 3.5 , 4 0.5 f 3.5 = 3 3.5 2 + 1 2 = 614.46875 0.5 × 614.46875 4 , 4.5 0.5 f 4 = 4 4 2 + 1 2 = 1156 0.5 × 1156 4.5 , 5 0.5 f 4.5 = 4.5 4.5 2 + 1 2 = 2032.03125 0.5 × 2032.03125 Therefore, for n = 8 , the left Riemann sum is, L 8 = 2 + 0.5 × 15.84375 + 0.5 × 50 + 0

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ISBN: 9780135165928

Author: BITTINGER

Publisher: PEARSON

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Chapter 4.7, Problem 9E

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To calculate: Ln,Rn and their average for given definite integral using the indicated value of n .

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Chapter 4.7, Problem 8E

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L n , R n and their average for given definite integral using the indicated value of n .

Solution Summary: The author calculates Ln, Rn and their average for given definite integral using the indicated value of n.

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To calculate: Ln,Rn and their average for given definite integral using the indicated value of n .

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Chapter 4 Solutions