(a) Explanation Given information: The integral function is ∫ 0 1 cos ( x 2 ) d x . The upper limit is 1 and lower limit is 0. The number of sub intervals as n = 8 . Calculation: Calculate the approximate value of integral by using Trapezoidal rule. Show the integral function as follows: ∫ 0 1 cos ( x 2 ) d x (1) Show the Trapezoidal rule as follows: ∫ a b f ( x ) d x ≈ T n = Δ x 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ... + 2 f ( x n − 1 ) + f ( x n ) ] Here, T n is trapezoidal approximation, x 1 , x 2 , ... x n are subintervals, Δ x = b − a n and x i = a + i Δ x . Calculate the length of the subinterval ( Δ x ) using the formula: Δ x = b − a n Here, b is upper limit, a is lower limit, and n is number of subintervals. Substitute 1 for b , 0 for a , and 8 for n . Δ x = 1 − 0 8 = 1 8 = 0.125 Hence, the length subinterval is 0.125 for the limits [ 0 , 1 ] . The subinterval values are 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, and 1 for x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , and x 8 . Apply Trapezoidal rule in Equation (1). ∫ 0 1 cos ( x 2 ) d x ≈ T n = Δ x 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ... + 2 f ( x n − 1 ) + f ( x n ) ] Substitute 0.125 for Δ x and 8 for n . T 8 = 0.125 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ... + 2 f ( x 7 ) + f ( x 8 ) ] (2) Apply the subinterval values in Equation (2). T 8 = 0.125 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ... + 2 f ( x 7 ) + f ( x 8 ) ] = 0.125 2 [ f ( 0 ) + 2 f ( 0.125 ) + 2 f ( 0.25 ) + 2 f ( 0.375 ) + 2 f ( 0.5 ) + 2 f ( 0.625 ) + 2 f ( 0.75 ) + 2 f ( 0.875 ) + f ( 1 ) ] (3) Apply f ( x ) = cos ( x 2 ) in Equation (3). T 8 = 0.125 2 [ f ( 0 ) + 2 f ( 0.125 ) + 2 f ( 0.25 ) + 2 f ( 0.375 ) + 2 f ( 0.5 ) + 2 f ( 0.625 ) + 2 f ( 0.75 ) + 2 f ( 0.875 ) + f ( 1 ) ] = 0.125 2 [ ( cos 0 2 ) + 2 ( cos 0.125 2 ) + 2 ( cos 0.25 2 ) + 2 ( cos 0.375 2 ) + 2 ( cos 0.5 2 ) + 2 ( cos 0.625 2 ) + 2 ( cos 0.75 2 ) + 2 ( cos 0.875 2 ) + ( cos 1 2 ) ] ≈ 0.902333 Therefore, the approximate value of integral by using Trapezoidal rule is 0 (b) To determine To Estimate : The error value in Trapezoidal rule and Midpoint rule for evaluation of the integral ∫ 0 1 cos ( x 2 ) d x . (c) To determine To Find: The larger value of n if guaranteed error value is within 0.0001 for Trapezoidal rule and Midpoint rule.

Bundle: Stewart, Essential Calculus: Early Transcendentals, 2nd (hardound) + WebAssign Printed Access Card for Stewart's Essential Calculus: Early ... Multi-Term + WebAssign - Start Smart Guide

2nd Edition

ISBN: 9781133425946

Author: James Stewart

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence t…

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

Videos

Question

Chapter 6.5, Problem 17E

(a)

To determine

To Calculate: the approximate value of the integral using Trapezoidal rule and Midpoint rule.

(b)

To determine

To Estimate: The error value in Trapezoidal rule and Midpoint rule for evaluation of the integral ∫01cos(x2)dx.

(c)

To determine

To Find: The larger value of n if guaranteed error value is within 0.0001 for Trapezoidal rule and Midpoint rule.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.5, Problem 16E

Chapter 6.5, Problem 18E

Students have asked these similar questions

Tips S ps L 50. lim x2 - 4 x-2x+2 51. lim 22 - X 52. 53. x 0 Answer lim x 0 lim 2-5 X 2x2 2 x² Answer -> 54. lim T - 3x - - 25 +5 b+1 b3b+3 55. lim X x-1 x 1 Answer 56. lim x+2 x 2 x 2 57. lim x²-x-6 x-2 x²+x-2 Answer-> 23-8 58. lim 2-22-2

S 36. lim 5x+2 x-2 37. lim √√2x4 + x² x-3 Answer-> 2x3 +4 38. lim x12 √ x² + 1 √√x² + 8 39. lim x-1 2x+4 Answer 40. lim x3 2x x√x² + 7 √√2x+3

See image

Chapter 6 Solutions

Bundle: Stewart, Essential Calculus: Early Transcendentals, 2nd (hardound) + WebAssign Printed Access Card for Stewart's Essential Calculus: Early ... Multi-Term + WebAssign - Start Smart Guide

Ch. 6.1 - Evaluate the integral using integration by parts...Ch. 6.1 - Evaluate the integral using integration by parts...Ch. 6.1 - Evaluate the integral. 3. xcos5xdx Ch. 6.1 - Evaluate the integral. 4. ye0.2ydy Ch. 6.1 - Evaluate the integral. 5. te3tdt Ch. 6.1 - Evaluate the integral. 6. (x1)sinxdx Ch. 6.1 - Evaluate the integral. 7. (x2+2x)cosxdx Ch. 6.1 - Evaluate the integral. 8. t2sintdt Ch. 6.1 - Evaluate the integral. ln(2x + 1) dx Ch. 6.1 - Evaluate the integral. p5lnpdp

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

the approximate value of the integral using Trapezoidal rule and Midpoint rule.

Solution Summary: The author calculates the approximate value of the integral using Trapezoidal rule and Midpoint rule.

Concept explainers

Videos

To Calculate: the approximate value of the integral using Trapezoidal rule and Midpoint rule.

To Estimate: The error value in Trapezoidal rule and Midpoint rule for evaluation of the integral ∫01cos(x2)dx.

To Find: The larger value of n if guaranteed error value is within 0.0001 for Trapezoidal rule and Midpoint rule.

Want to see the full answer?

Chapter 6 Solutions