(a) Explanation Given: The function f ( x ) = x 2 − 1 defined on interval [ 2 , 7 ] . 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 = 7 − 2 75 = 5 75 = 1 15 Then, the subintervals are as follows. [ 2 , 31 15 ] , [ 31 15 , 32 15 ] , [ 32 15 , 33 15 ] , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , [ 104 15 , 7 ] The left end points of the above subintervals are 2 , 31 15 , 32 15 , 33 15 , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , 104 15 and the right end points are 31 15 , 32 15 , 33 15 , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , 104 15 , 7 and the mid points are 61 30 , 63 30 , 65 30 , ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ , 209 30 . Computation of the left Riemann sum as follows. ∑ k = 1 75 f ( x k * ) Δ x = ∑ k = 1 75 f [ 2 + ( k − 1 ) Δ x ] Δ x [ ∵ x k * = a + ( k − 1 ) Δ x ] = ∑ k = 1 75 f [ 2 + ( k − 1 ) 1 15 ] × 1 15 = ∑ k = 1 75 [ ( 2 + ( k − 1 ) 1 15 ) 2 − 1 ] × 1 15 [ ∵ f ( x ) = x 2 − 1 ] = ∑ k = 0 74 [ ( 2 + ( k 15 ) ) 2 − 1 ] × 1 15 On further simplification, ∑ k = 1 75 f ( x k * ) Δ x = ∑ k = 0 74 [ ( 2 + ( k 15 ) ) 2 − 1 ] × 1 15 = [ ( 2 ) 2 + ( 2 + ( 1 15 ) ) 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ( 2 + ( 74 15 ) ) 2 ] × 1 15 − 75 15 = [ 4 + ( 31 15 ) 2 + ( 32 15 ) 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ( 104 15 ) 2 ] × 1 15 − 5 = 105.170 Computation of the right Riemann sum as follows. ∑ k = 1 75 f ( x k * ) Δ x = ∑ k = 1 75 f [ 2 + ( k ) Δ x ] Δ x [ ∵ x k * = a + ( k ) Δ x ] = ∑ k = 1 75 f [ 2 + ( k ) 1 15 ] × 1 15 = ∑ k = 1 75 [ ( 2 + ( k ) 1 15 ) 2 − 1 ] × 1 15 [ ∵ f ( x ) = x 2 − 1 ] = ∑ k = 1 75 [ ( 2 + ( k 15 ) ) 2 + 1 ] × 1 15 On further simplification, ∑ k = 1 75 f ( x k * ) Δ x = ∑ k = 1 75 [ ( 2 + ( k 15 ) ) 2 − 1 ] × 1 15 = [ ( 2 + ( 1 15 ) ) 2 + ( 2 + ( 2 15 ) ) 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ( 2 + ( 74 15 ) ) 2 + ( 2 + ( 75 15 ) ) 2 ] × 1 15 − 75 15 = [ ( 31 15 ) 2 + ( 32 15 ) 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ + ( 104 15 ) 2 + 7 2 ] × 1 15 − 5 = 108 (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 [ 2 , 7 ] .

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

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

Publisher: PEARSON

See similar textbooks

Question

Chapter 5.1, Problem 53E

(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=75 .

(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 [2,7] .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5.1, Problem 52E

Chapter 5.1, Problem 54E

Students have asked these similar questions

Complete the following steps for the given functionf and interval. a. For the given value of n, use sigma notation to write the left, right, and midpoint Riemann sums. Then evaluate each sum using a calculator. b. Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f on the interval. f(x) = /x for [0,20]; n= 40 a. Write the left Riemann sum. Choose the correct answer below. 40 1 k- O A. Σ k= 1 40 1 2k - 1 OB. 4. k= 1 40 1 k+1 Σ 40 k= 1 40 k= 1

Complete the following steps for the given function f and interval. a. For the given value of n, use sigma notation to write the left, right, and midpoint Riemann sums. Then evaluate each sum using a calculator. b. Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f on the interval. f(x) = cos 2x for 0,-; n=60 a. Write the left Riemann sum.

Complete the following steps for the given function f and interval. a. For the given value of n, use sigma notation to write the left, right, and midpoint Riemann sums. Then evaluate each sum using a calculator. b. Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f on the interval. f(x) = cos 2x for 0, n= 40 a. Write the left Riemann sum. 40 Σ k= 1 (Type an exact answer, using n as needed.)

Chapter 5 Solutions

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 5.1 - What is the displacement of an object that travels...Ch. 5.1 - Prob. 2QC Ch. 5.1 - If the interval [1, 9] is partitioned into 4...Ch. 5.1 - Prob. 4QC Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - The velocity in ft/s of an object moving along a...Ch. 5.1 - Prob. 6E

Knowledge Booster

The left Riemann sum, right Riemann sum and the midpoint Riemann sum in the form of sigma notation for n = 75 .

Solution Summary: The author explains the form of Riemann sum, sigma notation, and midpoint sum for n=75.

To write: The left Riemann sum, right Riemann sum and the midpoint Riemann sum in the form of sigma notation for n=75 .

To estimate: The area of the region bounded by the graph of the function f(x) and the x -axis on the interval [2,7] .

Want to see the full answer?

Chapter 5 Solutions