Explanation Given: The provided equation is: ∫ 1 4 ( 4 − x 2 ) d x And, the value n = 6 . Formula used: Trapezoidal rule is: ∫ a b f ( x ) d x ≈ b − a 2 n [ f ( x 0 ) + 2 f ( x 1 ) + .... + 2 f ( x n − 1 ) + f ( x n ) ] Simpson’s rule is: ∫ a b p ( x ) d x ≈ ( b − a 3 n ) [ 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 ) ] Calculation: Consider f ( x ) = 4 − x 2 . From the provided integral b = 4 , a = 1 and n = 6 . So, the width is: Δ x = b − a n = 4 − 1 6 = 3 6 = 1 2 Use trapezoidal rule: ∫ 1 4 ( 4 − x 2 ) d x ≈ 4 − 1 2 ( 6 ) [ f ( 1 + 1 2 × 0 ) + 2 f ( 1 + 1 2 × 1 ) + 2 f ( 1 + 1 2 × 2 ) + 2 f ( 1 + 1 2 × 3 ) + 2 f ( 1 + 1 2 × 4 ) + 2 f ( 1 + 1 2 × 5 ) + f ( 1 + 1 2 × 6 ) ] = 3 12 [ f ( 1 ) + 2 f ( 3 2 ) + 2 f ( 2 ) + 2 f ( 5 2 ) + 2 f ( 3 ) + 2 f ( 7 2 ) + f ( 4 ) ] Put the function and substitute the value of x : ∫ 1 4 ( 4 − x 2 ) d x = 1 4 [ ( 4 − ( 1 ) 2 ) + 2 ( 4 − ( 3 2 ) 2 ) + 2 ( 4 − ( 2 ) 2 ) + 2 ( 4 − ( 5 2 ) 2 ) + 2 ( 4 − ( 3 ) 2 ) + 2 ( 4 − ( 7 2 ) 2 ) + ( 4 − ( 4 ) 2 ) ] = 1 4 [ 3 + 2 ( 4 − 9 4 ) + 2 ( 4 − 4 ) + 2 ( 4 − 25 4 ) + 2 ( 4 − 9 ) + 2 ( 4 − 49 4 ) + ( 4 − 16 ) ] = 1 4 [ 3 + 2 × 7 4 + 2 × 0 + 2 × ( − 9 4 ) + 2 × ( − 5 ) + 2 ( − 33 4 ) + ( − 12 ) ] = 1 4 [ 3 + 3.5 + 0 − 4.5 − 10 − 16.5 − 12 ] Simplify the above equation as below: 1 4 [ 3 + 3

Calculus Of A Single Variable With Calcchat And Calcview, 11e

11th Edition

ISBN: 9781337275583

Author: Larson

Publisher: Cengage Learning

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Chapter 8.6, Problem 10E

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To calculate: The approximate value of definite integral for given value of n by Trapezoidal rule and Simpson’s rule and compare it with the exact value of the definite integral.

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Chapter 8.6, Problem 9E

Chapter 8.6, Problem 11E

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Chapter 8 Solutions

Calculus Of A Single Variable With Calcchat And Calcview, 11e

Ch. 8.1 - Integration Technique Describe how to integrate a...Ch. 8.1 - Fitting Integrands to Basic Integration Rules What...Ch. 8.1 - Choosing an Antiderivative In Exercises 3 and 4,...Ch. 8.1 - Choosing an Antiderivative In Exercises 3 and 4,...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...Ch. 8.1 - Choosing a Formula In Exercises 514, select the...

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The approximate value of definite integral for given value of n by Trapezoidal rule and Simpson’s rule and compare it with the exact value of the definite integral.

Solution Summary: The author compares the approximate value of the definite integral for n by Trapezoidal rule and Simpson’s rule.

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To calculate: The approximate value of definite integral for given value of n by Trapezoidal rule and Simpson’s rule and compare it with the exact value of the definite integral.

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Chapter 8 Solutions