Explanation Given Information: The definite integral is ∫ 1 3 ( 4 − x 2 ) d x , n = 4 . Formula used: 1. Trapezoidal Rule: If a function f is continuous on [ a , b ] , then ∫ a b f ( x ) d x ≈ ( b − a 2 n ) [ f ( x 0 ) + 2 f ( x 0 ) + ... + 2 f ( x n − 1 ) + f ( x n ) ] 2. Simpson’s Rule: If f is continuous on [ a , b ] and n is an even integer, then ∫ a b f ( x ) d x ≈ ( b − a 3 n ) [ f ( x 0 ) + 4 f ( x 1 ) + 2 f ( x 2 ) + 4 f ( x 3 ) + … + 4 f ( x n − 1 ) + f ( x n ) ] Calculation: Calculation to get exact value: Consider the definite integral ∫ 1 3 ( 4 − x 2 ) d x , n = 4 . ∫ 1 3 ( 4 − x 2 ) d x = ∫ 1 3 ( 4 x − x 3 3 ) = 4 ( 3 − 1 ) − ( 3 3 − 1 3 ) 3 = 4 ( 2 ) − ( 27 − 1 ) 3 Simplify as: ∫ 1 3 ( 4 − x 2 ) d x = 8 − 26 3 = 8 − 8.6666 = − 0.6666 ≈ − 0.6667 Therefore, the exact value is ∫ 1 3 ( 4 − x 2 ) d x ≈ − 0.6667 . Calculation by Trapezoidal Rule: Consider the definite integral ∫ 1 3 ( 4 − x 2 ) d x , n = 4 . When n = 4 , the width of each subinterval is, 3 − 1 4 = 2 4 = 1 2 And the end points of subintervals are, For x 0 , x 0 = 1 For x 1 , x 1 = 1 + 1 2 = 3 2 For x 2 , x 2 = 3 2 + 1 2 = 2 For x 3 , x 3 = 2 + 1 2 = 5 2 For x 4 , x 4 = 5 2 + 1 2 = 3 By Trapezoidal Rule, ∫

Calculus: An Applied Approach (Providence College: MTH 109)

9th Edition

ISBN: 9781285142616

Author: Ron Larson

Publisher: CENGAGE C

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Chapter 6.3, Problem 3E

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To calculate: The value of the integral ∫13(4−x2)dx,n=4 by use of the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the indicated value of n. Compare these results with the exact value of definite integral. Round your answers to four decimal places.

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Chapter 6.3, Problem 2E

Chapter 6.3, Problem 4E

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Chapter 6 Solutions

Calculus: An Applied Approach (Providence College: MTH 109)

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The value of the integral ∫ 1 3 ( 4 − x 2 ) d x , n = 4 by use of the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the indicated value of n . Compare these results with the exact value of definite integral. Round your answers to four decimal places.

Solution Summary: The author compares the results of exact value and Simpson's rule with the value obtained from Trapezoidal rule.

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To calculate: The value of the integral ∫13(4−x2)dx,n=4 by use of the Trapezoidal Rule and Simpson’s Rule to approximate the value of the definite integral for the indicated value of n. Compare these results with the exact value of definite integral. Round your answers to four decimal places.

Want to see the full answer?

Chapter 6 Solutions