(I) (a) To determine The integral with n = 4 steps using Trapezoidal rule and upper bound for | E T | . Explanation Given information: ∫ 2 4 1 ( s − 1 ) 2 d s Formula: Δ x = b − a n | E T | ≤ M ( b − a ) 3 12 n 2 Calculation: Show the actual value of definite integral as below: ∫ 2 4 1 ( s − 1 ) 2 d s (1) Show the definite integral f ( s ) as below: ∫ a b f ( s ) d s (2) Compare equation (1) and (2). f ( s ) = 1 ( s − 1 ) 2 Substitute 2 for s. f ( s 0 ) = 1 ( 2 − 1 ) 2 = 1 Similarly, calculate the remaining values of f ( s i ) for four equal subintervals from 2 to 4 and tabulate them in the 3 rd column of Table (1). Show the portion [ 2 , 4 ] divided into four equal subintervals and the values of f ( s i ) as in Table 1. s i f ( s i ) = y i s 0 1 1 s 1 5/2 4/9 s 2 3 1/4 s 3 7/2 4/25 s 4 4 1/9 Calculate the step size Δ s using the expression: Δ s = b − a n Substitute 4 for b, 2 for a and 4 for n. Δ s = 4 − 2 4 = 0.5 Evaluate the integral using the expression: T = Δ s 2 ( y 0 + 2 y 1 + 2 y 2 + 2 y 3 + y 4 ) Substitute 0.5 for Δ s , 1 for y 0 , 4 9 for y 1 , 1 4 for y 2 , 4 25 for y 3 and 1 9 for y 4 (b) To determine The value of the integral directly and its error. (c) To determine The percentage of the integral’s true value. (II) (a) To determine The integral with n = 4 steps using Simpson’s rule and upper bound for | E S | . (b) To determine The value of the integral directly and its error. (c) To determine The percentage of the integral’s true value.

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ISBN: 9780134665672

Author: Hass

Publisher: PEARSON

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Chapter 8.7, Problem 8E

(I)

(a)

To determine

The integral with n=4 steps using Trapezoidal rule and upper bound for |ET|.

(b)

To determine

The value of the integral directly and its error.

(c)

To determine

The percentage of the integral’s true value.

(II)

(a)

To determine

The integral with n=4 steps using Simpson’s rule and upper bound for |ES|.

(b)

To determine

The value of the integral directly and its error.

(c)

To determine

The percentage of the integral’s true value.

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Chapter 8.7, Problem 7E

Chapter 8.7, Problem 9E

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