Explanation Given: An integral is given as ∫ 0 4 x d x with number of subintervals n = 6 . 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 ∫ 0 4 x d x . Here, a = 0 , b = 4 , n = 6 Therefore, Δ x = b − a n = 4 − 0 6 = 4 6 = 2 3 Now, divide the interval [ 0 , 4 ] into n = 6 subintervals of length Δ x = 2 3 , with the following endpoints: a = 0 , 2 3 , 4 3 , 2 , 8 3 , 10 3 , 4 = b Evaluate the following functional values at these endpointsfor Trapezoidal’s Rule: f ( x 0 ) = f ( a ) = f ( 0 ) = 0 2 f ( x 1 ) = 2 f ( 2 3 ) = 2 ⋅ 2 3 = 1.63299316185545 2 f ( x 2 ) = 2 f ( 4 3 ) = 2 ⋅ 4 3 = 2.3094010767585 2 f ( x 3 ) = 2 f ( 2 ) = 2 ⋅ 2 = 2

6th Edition

ISBN: 9781465208880

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

Publisher: Kendall Hunt Publishing

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Chapter 5.8, Problem 14PS

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 13PS

Chapter 5.8, Problem 15PS

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Chapter 5 Solutions

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.

Solution Summary: The author calculates the value of the given integration by Simpson's Rule and trapezoidal rule for the specified number of intervals.

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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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