(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. Explanation Given information: ∫ 1 2 x d x Formula: Δ x = b − a n T = Δ x 2 ( y 0 + y 1 ) | E T | = ∫ 1 2 x d x − T Calculation: Show the actual value of definite integral as below: ∫ 1 2 x d x (1) Show the actual value of definite integral f ( x ) as below: ∫ a b f ( x ) d x (2) Compare equation (1) and (2). f ( x ) = x (3) Differentiate equation f ( x ) = x with respect to x. f ' ( x ) = 1 Again differentiate the above equation with respect to x. f ' ' ( x ) = 0 The value of | f ' ' | is equal to 0. The function 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 directly as below. ∫ 1 2 x d x = ( x 2 2 ) 1 2 = ( 2 2 2 − 1 2 2 ) = 2 − 1 2 = 3 2 Substitute 1 for x in equation (3) (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 plus MyLab Math with Pearson eText -- Title-Specific Access Card Package (14th Edition)

14th Edition

ISBN: 9780134768496

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

Publisher: PEARSON

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Chapter 8.7, Problem 11E

(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 10E

Chapter 8.7, Problem 12E

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