(a) Explanation Given information: The integral function is ∫ 0 1 cos ( x 2 ) d x . The upper limit is 1 and lower limit is 0. The number of sub intervals as n = 8 . Calculation: Calculate the approximate value of integral by using Trapezoidal rule. Show the integral function as follows: ∫ 0 1 cos ( x 2 ) d x (1) Show the Trapezoidal rule as follows: ∫ a b f ( x ) d x ≈ T n = Δ x 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ... + 2 f ( x n − 1 ) + f ( x n ) ] Here, T n is trapezoidal approximation, x 1 , x 2 , ... x n are subintervals, Δ x = b − a n and x i = a + i Δ x . Calculate the length of the subinterval ( Δ x ) using the formula: Δ x = b − a n Here, b is upper limit, a is lower limit, and n is number of subintervals. Substitute 1 for b , 0 for a , and 8 for n . Δ x = 1 − 0 8 = 1 8 = 0.125 Hence, the length subinterval is 0.125 for the limits [ 0 , 1 ] . The subinterval values are 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, and 1 for x 0 , x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , and x 8 . Apply Trapezoidal rule in Equation (1). ∫ 0 1 cos ( x 2 ) d x ≈ T n = Δ x 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ... + 2 f ( x n − 1 ) + f ( x n ) ] Substitute 0.125 for Δ x and 8 for n . T 8 = 0.125 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ... + 2 f ( x 7 ) + f ( x 8 ) ] (2) Apply the subinterval values in Equation (2). T 8 = 0.125 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ... + 2 f ( x 7 ) + f ( x 8 ) ] = 0.125 2 [ f ( 0 ) + 2 f ( 0.125 ) + 2 f ( 0.25 ) + 2 f ( 0.375 ) + 2 f ( 0.5 ) + 2 f ( 0.625 ) + 2 f ( 0.75 ) + 2 f ( 0.875 ) + f ( 1 ) ] (3) Apply f ( x ) = cos ( x 2 ) in Equation (3). T 8 = 0.125 2 [ f ( 0 ) + 2 f ( 0.125 ) + 2 f ( 0.25 ) + 2 f ( 0.375 ) + 2 f ( 0.5 ) + 2 f ( 0.625 ) + 2 f ( 0.75 ) + 2 f ( 0.875 ) + f ( 1 ) ] = 0.125 2 [ ( cos 0 2 ) + 2 ( cos 0.125 2 ) + 2 ( cos 0.25 2 ) + 2 ( cos 0.375 2 ) + 2 ( cos 0.5 2 ) + 2 ( cos 0.625 2 ) + 2 ( cos 0.75 2 ) + 2 ( cos 0.875 2 ) + ( cos 1 2 ) ] ≈ 0.902333 Therefore, the approximate value of integral by using Trapezoidal rule is 0 (b) To determine To Estimate : The error value in Trapezoidal rule and Midpoint rule for evaluation of the integral ∫ 0 1 cos ( x 2 ) d x . (c) To determine To Find: The larger value of n if guaranteed error value is within 0.0001 for Trapezoidal rule and Midpoint rule.

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ISBN: 9781133425946

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Chapter 6.5, Problem 17E

(a)

To determine

To Calculate: the approximate value of the integral using Trapezoidal rule and Midpoint rule.

(b)

To determine

To Estimate: The error value in Trapezoidal rule and Midpoint rule for evaluation of the integral ∫01cos(x2)dx.

(c)

To determine

To Find: The larger value of n if guaranteed error value is within 0.0001 for Trapezoidal rule and Midpoint rule.

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Chapter 6.5, Problem 16E

Chapter 6.5, Problem 18E

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Bundle: Stewart, Essential Calculus: Early Transcendentals, 2nd (hardound) + WebAssign Printed Access Card for Stewart's Essential Calculus: Early ... Multi-Term + WebAssign - Start Smart Guide

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the approximate value of the integral using Trapezoidal rule and Midpoint rule.

Solution Summary: The author calculates the approximate value of the integral using Trapezoidal rule and Midpoint rule.

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To Calculate: the approximate value of the integral using Trapezoidal rule and Midpoint rule.

To Estimate: The error value in Trapezoidal rule and Midpoint rule for evaluation of the integral ∫01cos(x2)dx.

To Find: The larger value of n if guaranteed error value is within 0.0001 for Trapezoidal rule and Midpoint rule.

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Chapter 6 Solutions