Explanation Given: An integral is given as ∫ 1 2 x 2 d x with number of subintervals n = 4 . Formula used: Trapezoidal rule to approximate integral is given by: ∫ a b f ( x ) d x ≈ Δ x 2 ( f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ... + 2 f ( x n − 1 ) + f ( x n ) ) Simpson’s rule to approximate integral is given by: ∫ a b f ( x ) d x ≈ Δ x 3 [ f ( x 0 ) + 4 ( f ( x 1 ) + f ( x 3 ) + ... + f ( x n − 1 ) ) + 2 ( f ( x 2 ) + f ( x 4 ) + ... + 2 f ( x n − 2 ) ) + f ( x n ) ] Where Δ x = b − a n Formula to evaluate the integral of the form ∫ a x n d x is shown below. ∫ a x n d x = a ∫ x n d x = a x n + 1 n + 1 + C , n ≠ − 1 Fundamental theorem of Calculus I: ∫ a b f ( x ) d x = F ( b ) − F ( a ) Where F ′ ( x ) = f ( x ) Calculation: The given integral is ∫ 1 2 x 2 d x . Here, a = 1 , b = 2 , n = 4 Therefore, Δ x = b − a n = 2 − 1 4 = 1 4 Now, divide the interval [ 1 , 2 ] into n = 4 subintervals of length Δ x = 1 4 , with the following endpoints: a = 1 , 5 4 , 3 2 , 7 4 , 2 = b Evaluate the following functional values at these endpointsfor Trapezoidal’s Rule: f ( x 0 ) = f ( a ) = f ( 1 ) = 1 2 f ( x 1

Student's Solution and Survival Manual for Calculus

7th Edition

ISBN: 9781524934040

Author: STRAUSS MONTY J, TODA MAGDALENA DANIELE, SMITH KARL J

Publisher: Kendall Hunt Publishing

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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…

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Chapter 5.8, Problem 12PS

To determine

To calculate: The value of the given integration by Simpson’s Rule and trapezoidal rule for the specified number of intervals and compare with the actual value of the integral.

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Chapter 5.8, Problem 11PS

Chapter 5.8, Problem 13PS

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Chapter 5 Solutions

Student's Solution and Survival Manual for Calculus

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The value of the given integration by Simpson’s Rule and trapezoidal rule for the specified number of intervals and compare with the actual value of the integral.

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To calculate: The value of the given integration by Simpson’s Rule and trapezoidal rule for the specified number of intervals and compare with the actual value of the integral.

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Chapter 5 Solutions