Explanation Given information: ∫ 0 1 ( x − 1 2 ) 2 d x with n = 4 , where n is the number of equal subintervals Formula used: Trapezoidal rule: ∫ a b f ( x ) d x ≈ [ f ( a 0 ) + 2 f ( a 1 ) + ... + 2 f ( a n − 1 ) + f ( a n ) ] Δ x 2 Where a 0 , a 1 , ... , a n are end points of the subintervals a ≤ x ≤ b and Δ x = b − a n Calculation: For n = 4 Compare ∫ a b f ( x ) d x to ∫ 0 1 ( x − 1 2 ) 2 d x ; it becomes a = 0 and b = 1 Therefore, Δ x = b − a n = 1 − 0 4 = 0.25 Now, a 0 = a ⇒ a 0 = 0 a 1 = a 0 + Δ x = 0 + 0.25 = 0.25 a 2 = a 1 + Δ x = 0.25 + 0.25 = 0.50 a 3 = a 2 + Δ x = 0.5 + 0.25 = 0.75 a 4 = a 3 + Δ x = 0.75 + 0.25 = 1 By using trapezoidal rule, ∫ 0 1 ( x − 1 2 ) 2 d x ≈ [ f ( 0 ) + 2 f ( 0.25 ) + 2 f ( 0.5 ) + 2 ( 0.75 ) + f ( 1 ) ] × ( 0.25 2 ) ⇒ ∫ 0 1 ( x − 1 2 ) 2 d x ≈ [ ( 0 − 1 2 ) 2 + 2 ( 0

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Numerical Integration

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Chapter 9.4, Problem 11E

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To calculate: The approximate value of the integral ∫01(x−12)2dx by trapezoidal rule with n=4 andexact value of integral ∫01(x−12)2dx. Express the answer to five decimal places.

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Chapter 9.4, Problem 10E

Chapter 9.4, Problem 12E

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The approximate value of the integral ∫ 0 1 ( x − 1 2 ) 2 d x by trapezoidal rule with n = 4 andexact value of integral ∫ 0 1 ( x − 1 2 ) 2 d x . Express the answer to five decimal places.

Solution Summary: The author calculates the approximate value of the integral by trapezoidal rule, where n is the number of equal subintervals.

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To calculate: The approximate value of the integral ∫01(x−12)2dx by trapezoidal rule with n=4 andexact value of integral ∫01(x−12)2dx. Express the answer to five decimal places.

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Chapter 9 Solutions