Explanation Given: The function f ( x ) = 3 x 2 + 1 defined on interval [ − 1 , 1 ] . Formula used: 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 . Calculation: Divide the interval [ − 1 , 1 ] into n equal subintervals with length, Δ x = b − a n = 1 + 1 n = 2 n . Compute the area by using right Riemann sum as follows. A n = ∑ k = 1 n f ( x k * ) Δ x = ∑ k = 1 n f [ − 1 + ( k ) Δ x ] Δ x [ ∵ x k * = a + ( k ) Δ x ] = ∑ k = 1 n f [ − 1 + ( k ) ( 2 n ) ] × 2 n = ∑ k = 1 n [ 3 ( − 1 + ( k ) ( 2 n ) ) 2 + 1 ] × 2 n [ ∵ f ( x ) = 3 x 2 + 1 ] Now, for n = 10 subintervals the area of the region is as follows. A 10 = ∑ k = 1 10 [ 3 ( − 1 + ( k ) ( 2 10 ) ) 2 + 1 ] × 2 10 = 3 [ ( − 1 + 2 10 ) 2 + ( − 1 + 4 10 ) 2 + ( − 1 + 6 10 ) 2 + ⋯ + ( − 1 + 20 10 ) 2 ] × 2 10 + 2 10 × 10 = 3 [ ( − 8 10 ) 2 + ( − 6 10 ) 2 + ( − 4 10 ) 2 + ⋯ + ( − 10 10 ) 2 ] × 2 10 + 2 10 × 10 ≈ 4.04 For n = 30 subintervals the area of the region is as follows. A 30 = ∑ k = 1 30 [ 3 ( − 1 + ( k ) ( 2 30 ) ) 2 + 1 ] × 2 30 = 3 [ ( − 1 + 2 30 ) 2 + ( − 1 + 4 30 ) 2 + ( − 1 + 6 30 ) 2 + ⋯ + ( − 1 + 60 30 ) 2 ] × 2 30 + 2 30 × 30 = 3 [ ( − 28 30 ) 2 + ( − 26 30 ) 2 + ( − 24 30 ) 2 + ⋯ + ( − 30 30 ) 2 ] × 2 30 + 2 30 × 30 ≈ 4

Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134763644

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

Publisher: PEARSON

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Chapter 5.1, Problem 56E

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To approximate: The area of the region bounded by the function f(x)=3x2+1 and the x-axis over the interval [−1,1] with n=10, 30, 60, 80 subintervals by using the right Riemann sum and make a conjecture about the limit of Riemann sum as n→∞.

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Chapter 5.1, Problem 55E

Chapter 5.1, Problem 57E

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Chapter 5 Solutions

Calculus: Early Transcendentals (3rd Edition)

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To approximate: The area of the region bounded by the function f ( x ) = 3 x 2 + 1 and the x -axis over the interval [ − 1 , 1 ] with n = 10 , 30 , 60 , 80 subintervals by using the right Riemann sum and make a conjecture about the limit of Riemann sum as n → ∞ .

To approximate: The area of the region bounded by the function f(x)=3x2+1 and the x-axis over the interval [−1,1] with n=10, 30, 60, 80 subintervals by using the right Riemann sum and make a conjecture about the limit of Riemann sum as n→∞.

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