(a) Explanation Given: The function f ( x ) = x 2 + 1 defined on interval [ − 1 , 1 ] . Formula used: (1). The left Riemann sum for the function f ( x ) defined on the interval [ a , b ] for n subinterval is ∑ k = 1 n f ( x k * ) Δ x = ∑ k = 1 n f [ a + ( k − 1 ) Δ x ] Δ x . (2). The right Riemann sum for the function f ( x ) defined on the interval [ a , b ] for n subinterval is ∑ k = 1 n f ( x k * ) Δ x = ∑ k = 1 n f [ a + ( k ) Δ x ] Δ x . (3). The midpoint Riemann sum for the function f ( x ) defined on the interval [ a , b ] for n subinterval is ∑ k = 1 n f ( x k * ) Δ x = ∑ k = 1 n f [ a + ( k − 1 2 ) Δ x ] Δ x Calculation: First find the length of the each subinterval, Δ x = 1 + 1 50 = 1 25 Then, the subintervals are, [ − 1 , − 24 25 ] , [ − 24 25 , − 23 25 ] , [ − 23 25 , − 22 25 ] , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , [ 24 25 , 1 ] . The left end points of the above subintervals are − 1 , − 24 25 , − 23 25 , − 22 25 , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , 24 25 and the right end points are − 24 25 , − 23 25 , − 22 25 , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , 1 and the mid points are − 49 50 , − 47 50 , − 45 50 , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , 49 50 . Computation of the left Riemann sum as follows. ∑ k = 1 50 f ( x k * ) Δ x = ∑ k = 1 50 f [ − 1 + ( k − 1 ) Δ x ] Δ x [ ∵ x k * = a + ( k − 1 ) Δ x ] = ∑ k = 1 50 f [ − 1 + ( k − 1 ) 1 25 ] × 1 25 = ∑ k = 1 50 [ ( − 1 + ( k − 1 ) 1 25 ) 2 + 1 ] × 1 25 [ ∵ f ( x ) = x 2 + 1 ] = ∑ k = 0 49 [ ( − 1 + ( k 25 ) ) 2 + 1 ] × 1 25 On further simplification, ∑ k = 1 50 f ( x k * ) Δ x = ∑ k = 0 49 [ ( − 1 + ( k 25 ) ) 2 + 1 ] × 1 25 = 50 25 + [ ( − 1 ) 2 + ( − 1 + ( 1 25 ) ) 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ( − 1 + ( 49 25 ) ) 2 ] × 1 25 = 2 + [ 1 + ( − 24 25 ) 2 + ( − 23 25 ) 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ( 24 25 ) 2 ] × 1 25 2.667 Computation of the right Riemann sum as follows (b) To determine To estimate: The area of the region bounded by the graph of the function f ( x ) and the x -axis on the interval [ − 1 , 1 ] .

Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book

2nd Edition

ISBN: 9780321954237

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Question

Chapter 5.1, Problem 44E

(a)

To determine

To write: The left Riemann sum, right Riemann sum and the midpoint Riemann sum in the form of sigma notation for n=50 .

(b)

To determine

To estimate: The area of the region bounded by the graph of the function f(x) and the x -axis on the interval [−1,1] .

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Chapter 5.1, Problem 43E

Chapter 5.1, Problem 45E

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The left Riemann sum, right Riemann sum and the midpoint Riemann sum in the form of sigma notation for n = 50 .

Solution Summary: The author explains how the left Riemann sum, the right, and the midpoint are in the form of sigma notation for n=50.

To write: The left Riemann sum, right Riemann sum and the midpoint Riemann sum in the form of sigma notation for n=50 .

To estimate: The area of the region bounded by the graph of the function f(x) and the x -axis on the interval [−1,1] .

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Chapter 5 Solutions

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