(a) Explanation Given: The integral function is ∫ 0 1 sin ( x 3 ) d x . The number of sub intervals as n = 10 . Formula used: ∫ 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 ) ] where, 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 . The length of the subinterval ( Δ x ) is given by the formula Δ x = b − a n where, b is upper limit, a is lower limit, and n is number of subintervals. Calculation: The given integral function is ∫ 0 1 sin ( x 3 ) d x ⋯ ⋯ ( 1 ) Thus, f ( x ) = sin ( x 3 ) and a = 0 , b = 1 and n = 10 . Substitute a = 0 , b = 1 and n = 10 and calculate the length of the subinterval ( Δ x ) as, Δ x = 1 − 0 10 = 1 10 = 0.1 Hence, the length of subinterval is 0.1 for the limits [ 0 , 1 ] . That is, the subinterval values are x 0 = 0 , x 1 = 0.1 , x 2 = 0.2 , x 3 = 0.3 , x 4 = 0.4 , x 5 = 0.5 , x 6 = 0.6 , x 7 = 0.7 , x 8 = 0.8 , x 9 = 0.9 and x 10 = 1 . Apply Trapezoidal rule and obtain the value of the given integral as shown below: Substitute f ( x ) = sin ( x 3 ) , Δ x = 0.1 and n = 10 in the above formula as shown below: ∫ 0 1 sin ( x 3 ) d x ≈ T 10 = Δ x n [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + ... + 2 f ( x n − 1 ) + f ( x n ) ] = 0.1 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + (b) To determine Calculate the approximate value of integral by using Midpoint rule. (c) To determine Calculate the approximate value of integral by using Simpson’s rule.

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Author: Stewart

Publisher: VST

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Chapter 6.5, Problem 12E

(a)

To determine

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

(b)

To determine

Calculate the approximate value of integral by using Midpoint rule.

(c)

To determine

Calculate the approximate value of integral by using Simpson’s rule.

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Chapter 6.5, Problem 11E

Chapter 6.5, Problem 13E

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Chapter 6 Solutions

EBK WEBASSIGN FOR STEWART'S ESSENTIAL C

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The approximate value of the integral using Trapezoidal rule.

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

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

Calculate the approximate value of integral by using Midpoint rule.

Calculate the approximate value of integral by using Simpson’s rule.

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Chapter 6 Solutions