Explanation Given information: ∫ 0 3 x 4 − x d x With n = 5 . where n is number of equal subintervals. Formula used: 1) Midpoint Rule: M = ∫ a b f ( x ) d x ≈ [ f ( x 1 ) + f ( x 2 ) + ⋯ + f ( x n ) ] Δ x Where, x 1 , x 2 , ... , x n are mid-points of the subintervals of the intervals a ≤ x ≤ b and Δ x = b − a n . 2) Trapezoidal Rule: T = ∫ a b f ( x ) d x ≈ [ f ( a 0 ) + 2 f ( a 1 ) + ... + 2 f ( a n − 1 ) + f ( a n ) ] Δ x 2 Where, a 0 , a 1 , ... , a n are end points of subintervals a ≤ x ≤ b and Δ x = b − a n . 3) Simpson’s Rule: ∫ a b f ( x ) ≈ S = 2 M + T 3 . Where, M the approximate value of integral is calculated by midpoint rule and T is the approximate value of integral calculated by Trapezoidal rule. 4) Integration by parts: ∫ f ( x ) g ( x ) d x = f ( x ) G ( x ) − ∫ f ′ ( x ) G ( x ) d x Where, f ' ( x ) is derivative of f ( x ) and G ( x ) is an antiderivative of g ( x ) . Calculation: For n = 5 Compare ∫ a b f ( x ) d x with ∫ 0 3 x 4 − x d x it becomes a = 0 and b = 3 Therefore, Δ x = b − a n = 3 − 0 5 = 0.6 Now, a 0 = a ⇒ a 0 = 0 x 1 = a 0 + Δ x 2 = 0 + 0.6 2 = 0.3 x 2 = x 1 + Δ x = 0.3 + 0.6 = 0.9 x 3 = x 2 + Δ x = 0.9 + 0.6 = 1.5 x 4 = x 3 + Δ x = 1.5 + 0.6 = 2.1 x 5 = x 4 + Δ x = 2.1 + 0.6 = 2.7 By using mid-point rule, M = ∫ 0 3 x 4 − x d x = [ f ( 0.3 ) + f ( 0.9 ) + f ( 1.5 ) + f ( 2.1 ) + f ( 2.7 ) ] × 0.6 = [ 0.3 4 − 0.3 + 0.9 4 − 0.9 + 1.5 4 − 1.5 + 2.1 4 − 2.1 + 2.7 4 − 2.7 ] × 0.6 ≈ 6.30390 ⇒ M = ∫ 0 3 x 4 − x d x ≈ 6.30390 Also, a 0 = a ⇒ a 0 = 0 a 1 = a 0 + Δ x = 0 + 0.6 = 0.6 a 2 = a 1 + Δ x = 0

Calculus & Its Applications (14th Edition)

14th Edition

ISBN: 9780134437774

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar

Publisher: PEARSON

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 9.4, Problem 18E

To determine

To calculate: The approximate value of integral ∫03x4−xdx by midpoint rule, trapezoidal rule, Simson’s rule with n=5 and find the exact value of integration. Express answers to five decimal places.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.4, Problem 17E

Chapter 9.4, Problem 19E

Students have asked these similar questions

Directions: Calculate the following definite integral. Use the substitution rule or integration by parts, if necessary. 3 [³ (9x + 8)e²/º da

Q5. Use integration by part to evaluate ſešx sin (e4x )dx

Chapter 9 Solutions

Calculus & Its Applications (14th Edition)

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 integral ∫ 0 3 x 4 − x d x by midpoint rule, trapezoidal rule, Simson’s rule with n = 5 and find the exact value of integration . Express answers to five decimal places.

Solution Summary: The author calculates the approximate value of integral by midpoint rule, trapezoidal rule and Simson’s rule with n=5.

Concept explainers

Videos

To calculate: The approximate value of integral ∫03x4−xdx by midpoint rule, trapezoidal rule, Simson’s rule with n=5 and find the exact value of integration. Express answers to five decimal places.

Want to see the full answer?

Chapter 9 Solutions