Explanation Given information: The curve function is y = sin x (1) The lower limit is a = 0 and the upper limit is b = π . The number of intervals is 10. Calculation: The Simpson’s rule is shown below: ∫ a b f ( x ) d x ≈ S n = Δ x 3 [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + ... + 2 f ( x n − 2 ) + 4 f ( x n − 1 ) + f ( x n ) ] (2) The interval width Δ x = b − a n (3) Here, the value of the function f ( x ) at x = 0 to n is f ( x 0 ) to f ( x n ) , the upper limit is b , the lower limit is a , and the number of interval is n . Find the value of f ( x ) using the relation: f ( x ) = 1 + ( d y d x ) 2 (4) Here, the derivative of the function y is d y d x . Differentiate Equation (1) with respect to x . d y d x = cos x Substitute cos x for d y d x in Equation (4). f ( x ) = 1 + cos 2 x (5) Find the width of the interval using Equation (3): Substitute π for b , 0 for a , and 10 for n in Equation (3). Δ x = π − 0 10 = π 10 Modify Simpson’s rule for n = 10 . S 10 = Δ x 3 [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + ... + 2 f ( x 8 ) + 4 f ( x 9 ) + f ( x 10 ) ] (6) Split the interval 0 to π into 10 subintervals. Thus the intervals are 0 , π 10 , 2 π 10 , 3 π 10 , 4 π 10 , 5 π 10 , 6 π 10 , 7 π 10 , 8 π 10 , 9 π 10 , and π . Substitute 0 for x 0 , π 10 for x 1 , 2 π 10 for x 2 , 3 π 10 for x 3 , 4 π 10 for x 4 , 5 π 10 for x 5 , 6 π 10 for x 6 , 7 π 10 for x 7 , 8 π 10 for x 8 , 9 π 10 for x 9 , and π for x 10 in Equation (6)

8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differ…

Videos

Question

Chapter 8, Problem 7RE

To determine

To find: The length of the curve using Simpson’s rule.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8, Problem 6RE

Chapter 8, Problem 8RE

Students have asked these similar questions

10 The hypotenuse of a right triangle has one end at the origin and one end on the curve y = Express the area of the triangle as a function of x. A(x) =

In Problems 17-26, solve the initial value problem. 17. dy = (1+ y²) tan x, y(0) = √√3

could you explain this as well as disproving each wrong option

Chapter 8 Solutions

Single Variable Calculus

Ch. 8.1 - Use the arc length formula (3) to find the length...Ch. 8.1 - Use the arc length formula to find the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Find the exact length of the curve. 9. y = 1 +...Ch. 8.1 - Find the exact length of the curve. 10. 36y2 = (x2...

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 length of the curve using Simpson’s rule.

Solution Summary: The author explains the length of the curve by using Simpson's rule.

Concept explainers

Videos

To find: The length of the curve using Simpson’s rule.

Want to see the full answer?

Chapter 8 Solutions