(I) (a) To determine The integral with n = 4 steps using the Trapezoidal rule and upper bound for | E T | . Explanation Given information: ∫ 0 π sin t d t 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: ∫ 0 π sin t d t (1) Show the definite integral f ( t ) as below: ∫ a b f ( t ) d t (2) Compare equation (1) and (2). f ( t ) = sin t Substitute 0 for t. f ( t 0 ) = sin 0 = 0 Similarly, calculate the remaining values of f ( t i ) for four equal subintervals from 0 to π and tabulate them in the 3 rd column of Table (1). Show the portion [ π , 0 ] divided into four equal subintervals and the values of f ( t i ) as in Table 1. t i f ( t i ) = y i t 0 0 0 t 1 π / 4 2 / 2 t 2 π / 2 1 t 3 3 π / 4 2 / 2 t 4 π 0 Calculate the step size Δ t using the expression: Δ t = b − a n Substitute π for b, 0 for a and 4 for n. Δ t = π − 0 4 = π 4 Evaluate the integral using the expression: T = Δ t 2 ( y 0 + 2 y 1 + 2 y 2 + 2 y 3 + y 4 ) Substitute π 4 for Δ t , 0 for y 0 , 2 2 for y 1 , 1 for y 2 , 2 2 for y 3 and 0 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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Chapter 8.7, Problem 9E

(I)

(a)

To determine

The integral with n=4 steps using the 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 8E

Chapter 8.7, Problem 10E

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