Explanation Given information: ∫ − 1 1 1 1 + x 2 d x with n = 5 , where n is the number of equal subintervals. Formula used: Midpoint Rule: ∫ a b f ( x ) d x ≈ f ( x 1 ) Δ x + f ( x 2 ) Δ x + ⋯ + f ( x n ) Δ x = [ f ( x 1 ) + f ( x 2 ) + ⋯ + f ( x n ) ] Δ x 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 The trapezoidal rule uses the values of f ( x ) at the endpoints of the interval a ≤ a i ≤ b Simpson’s Rule: S = 2 M + T 3 Where M the approximate value of the integral is calculated by midpoint rule and T is the approximate value of integral calculated by Trapezoidal rule. Calculation: Compare ∫ a b f ( x ) d x with ∫ − 1 1 1 1 + x 2 d x , it becomes a = − 1 and b = 1 . Therefore, Δ x = b − a n = 1 − ( − 1 ) 5 = 0.4 . The endpoints of the intervals begin at a = − 1 and are spaced 0.4 units apart. The first midpoint is at a + Δ x 2 = − 1 + 0.4 2 = − 0.8 . ⇒ x 1 = − 0.8 The midpoints are also spaced 0.4 unit apart. Therefore x 2 = − 0.4 , x 3 = 0 , x 4 = 0.4 and x 5 = 0.8 . By using the midpoint rule, the integral is approximately equal to ∫ − 1 1 1 1 + x 2 d x ≈ [ 1 1 + ( − 0.8 ) 2 + 1 1 + ( − 0.4 ) 2 + 1 1 + ( 0 ) 2 + 1 1 + ( 0.4 ) 2 + 1 1 + ( 0.8 ) 2 ] ( 0.4 ) ∫ − 1 1 1 1 + x 2 d x ≈ [ 1 1

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Author: Larry Goldstein / David Lay

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Chapter 9, Problem 46RE

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To calculate: The approximate value of the integral ∫−1111+x2dx by midpoint rule, trapezoidal

rule, Simson’s rule with n=5, where n is the number of equal subintervals.

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Chapter 9, Problem 45RE

Chapter 9, Problem 47RE

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The approximate value of the integral ∫ − 1 1 1 1 + x 2 d x by midpoint rule, trapezoidal rule, Simson’s rule with n = 5 , where n is the number of equal subintervals.

Solution Summary: The author calculates the approximate value of the integral 'displaystyle' by the midpoint rule, trapezoidal rule and Simson's rule.

Concept explainers

Videos

To calculate: The approximate value of the integral ∫−1111+x2dx by midpoint rule, trapezoidal

rule, Simson’s rule with n=5, where n is the number of equal subintervals.

Want to see the full answer?

Chapter 9 Solutions

The approximate value of the integral ∫ − 1 1 1 1 + x 2 d x by midpoint rule, trapezoidal rule, Simson’s rule with n = 5 , where n is the number of equal subintervals.

Solution Summary: The author calculates the approximate value of the integral 'displaystyle' by the midpoint rule, trapezoidal rule and Simson's rule.

Concept explainers

Videos

To calculate: The approximate value of the integral ∫−1111+x2dx by midpoint rule, trapezoidal rule, Simson’s rule with n=5, where n is the number of equal subintervals.

Want to see the full answer?

Chapter 9 Solutions

To calculate: The approximate value of the integral ∫−1111+x2dx by midpoint rule, trapezoidal

rule, Simson’s rule with n=5, where n is the number of equal subintervals.