(I) (a) To determine The integral with n = 4 steps using Trapezoidal rule and upper bound for | E T | . Explanation Given information: ∫ − 2 0 ( x 2 − 1 ) d x 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 0 ( x 2 − 1 ) d x (1) Show the definite integral f ( x ) as below: ∫ a b f ( x ) d x (2) Compare equation (1) and (2). f ( x ) = ( x 2 − 1 ) Substitute -2 for x. f ( x 0 ) = ( − 2 ) 2 − 1 = 3 Similarly, calculate the remaining values of f ( x i ) for four equal subintervals from -2 to 0 and tabulate them in the 3 rd column of Table (1). Show the portion [ − 2 , 0 ] divided into four equal subintervals and the values of f ( x i ) as in Table 1. x i f ( x i ) = y i x 0 -2 3 x 1 -3/2 5/4 x 2 -1 0 x 3 -1/2 -3/4 x 4 0 -1 Calculate the step size Δ x using the expression: Δ x = b − a n Substitute 0 for b, -2 for a and 4 for n. Δ x = 0 − ( − 2 ) 4 = 2 4 = 0.5 Evaluate the integral using the expression: T = Δ x 2 ( y 0 + 2 y 1 + 2 y 2 + 2 y 3 + y 4 ) Substitute 0 (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.

14th Edition

ISBN: 9780134438986

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

Publisher: PEARSON

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Question

Chapter 8.7, Problem 4E

(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 3E

Chapter 8.7, Problem 5E

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