(a) Explanation Given information: ∫ 1 3 ( 2 x − 1 ) d x Formula: Δ x = b − a n T = Δ x 2 ( y 0 + y 1 ) | E T | = ∫ 1 3 ( 2 x − 1 ) d x − T Calculation: Show the actual value of definite integral as below: ∫ 1 3 ( 2 x − 1 ) d x (1) Show the definite integral f ( x ) as below: ∫ a b f ( x ) d x (2) Differentiate equation f ( x ) = 2 x − 1 with respect to x. f ' ( x ) = 2 Again differentiate the above equation with respect to x. f ' ' ( x ) = 0 The value of | f ' ' | is equal to 0. The term f ' ' is continuous and M is an upper bound for the values of | f ' ' | on [ a , b ] . Thus, the value of M becomes 0. Thus, it is recommended to estimate the minimum number of sub intervals by means of trial and error method. Calculate the exact value of the integral as given below. ∫ 1 3 ( 2 x − 1 ) d x = ( 2 x 2 2 − x ) 1 3 = 3 2 − 3 − ( 1 2 − 1 ) = 9 − 3 − 1 + 1 = 6 Compare equation (1) and (2). f ( x ) = 2 x − 1 (3) Substitute 1 for x (b) To determine The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10 − 4 by Simpson’s rule.

Thomas' Calculus: Early Transcendentals in SI Units

14th Edition

ISBN: 9781292253114

Author: Hass, Joel R., Heil, Christopher E., WEIR, Maurice D.

Publisher: PEARSON

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Chapter 8.7, Problem 12E

(a)

To determine

The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10−4 by Trapezoidal rule.

(b)

To determine

The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10−4 by Simpson’s rule.

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Chapter 8.7, Problem 11E

Chapter 8.7, Problem 13E

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Thomas' Calculus: Early Transcendentals in SI Units

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The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10 − 4 by Trapezoidal rule.

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The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10−4 by Trapezoidal rule.

The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10−4 by Simpson’s rule.

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