Explanation Given information: ∫ 1 4 e x x + 1 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 the Trapezoidal rule. Calculation: Compare ∫ a b f ( x ) d x with ∫ 1 4 e x x + 1 d x , it becomes a = 1 and b = 4 . Therefore, Δ x = b − a n = 4 − 1 5 = 0.6 . The endpoints of the intervals begin at a = 1 and are spaced 0.6 units apart. The first midpoint is at a + Δ x 2 = 1 + 0.6 2 = 1.3 . ⇒ x 1 = 1.3 The midpoints are also spaced 0.6 units apart. Therefore x 2 = 1.9 , x 3 = 2.5 , x 4 = 3.1 and x 5 = 3.7 . By using the midpoint rule, the integral is approximately equal to ∫ 1 4 e x x + 1 d x ≈ [ e ( 1.3 ) 1.3 + 1 + e ( 1.9 ) 1.9 + 1 + e ( 2.5 ) 2.5 + 1 + e ( 3.1 ) 3.1 + 1 + e ( 3.7 ) 3.7 + 1 ] ( 0.6 ) ∫ 1 4 e x x + 1 d x ≈ [ e ( 1

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Chapter 9, Problem 45RE

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To calculate: The approximate value of the integral ∫14exx+1dx by midpoint rule, trapezoidal rule,

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

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Chapter 9, Problem 44RE

Chapter 9, Problem 46RE

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The approximate value of the integral ∫ 1 4 e x x + 1 d x by midpoint rule, trapezoidal rule, Simpson’s rule with n = 5 where n is the number of equal subintervals.

Solution Summary: The author calculates the approximate value of the integral by midpoint rule, trapezoidal rule and Simpson's rule with n=5.

Concept explainers

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To calculate: The approximate value of the integral ∫14exx+1dx by midpoint rule, trapezoidal rule,

Simpson’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 4 e x x + 1 d x by midpoint rule, trapezoidal rule, Simpson’s rule with n = 5 where n is the number of equal subintervals.

Solution Summary: The author calculates the approximate value of the integral by midpoint rule, trapezoidal rule and Simpson's rule with n=5.

Concept explainers

Videos

To calculate: The approximate value of the integral ∫14exx+1dx by midpoint rule, trapezoidal rule, Simpson’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 ∫14exx+1dx by midpoint rule, trapezoidal rule,

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