Explanation Given information: ∫ 1 4 ( 2 x − 3 ) 3 d x with n = 3 . Where, n is number of equal subintervals. Formula used: 1) Midpoint Rule: M = ∫ a b f ( x ) d x ≈ [ f ( x 1 ) + f ( x 2 ) + ⋯ + f ( x n ) ] Δ x Where, x 1 , x 2 , ... , x n are mid-points of the subintervals of the intervals a ≤ x ≤ b and Δ x = b − a n . 2) Trapezoidal Rule: T = ∫ 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 subintervals a ≤ x ≤ b and Δ x = b − a n . 3) Simpson’s Rule: ∫ a b f ( x ) ≈ S = 2 M + T 3 . Where, M the approximate value of integral is calculated by midpoint rule and T is the approximate value of integral calculated by Trapezoidal rule. 4) If f is a continuous function on an interval [ a , b ] with an antiderivative F then F ′ ( x ) = f ( x ) and ∫ a b f ( x ) d x = F ( b ) − F ( a ) Calculation: For n = 3 Compare ∫ a b f ( x ) d x with ∫ 1 4 ( 2 x − 3 ) 3 d x it becomes a = 1 and b = 4 Therefore, Δ x = b − a n = 4 − 1 3 = 1 Now, a 0 = a ⇒ a 0 = 1 x 1 = a 0 + Δ x 2 = 1 + 1 2 = 1.5 x 2 = x 1 + Δ x = 1.5 + 1 = 2.5 x 3 = x 2 + Δ x = 2.5 + 1 = 3.5 By using mid-point rule, M = ∫ 1 4 ( 2 x − 3 ) 3 d x = [ f ( 1.5 ) + f ( 2.5 ) + f ( 3.5 ) ] × 1 = [ ( 2 × 1

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Chapter 9.4, Problem 15E

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To calculate: The approximate value of integral ∫14(2x−3)3dx by midpoint rule, trapezoidal rule,

Simson’s rule with n=3 and find the exact value of integration. Express answers to five decimal places.

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Chapter 9.4, Problem 14E

Chapter 9.4, Problem 16E

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The approximate value of integral ∫ 1 4 ( 2 x − 3 ) 3 d x by midpoint rule, trapezoidal rule, Simson’s rule with n = 3 and find the exact value of integration. Express answers to five decimal places.

Solution Summary: The author calculates the approximate value of integral by midpoint rule, trapezoidal rule and Simson’s rule with n=3.

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To calculate: The approximate value of integral ∫14(2x−3)3dx by midpoint rule, trapezoidal rule,

Simson’s rule with n=3 and find the exact value of integration. Express answers to five decimal places.

Want to see the full answer?

Chapter 9 Solutions

The approximate value of integral ∫ 1 4 ( 2 x − 3 ) 3 d x by midpoint rule, trapezoidal rule, Simson’s rule with n = 3 and find the exact value of integration. Express answers to five decimal places.

Solution Summary: The author calculates the approximate value of integral by midpoint rule, trapezoidal rule and Simson’s rule with n=3.

Concept explainers

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To calculate: The approximate value of integral ∫14(2x−3)3dx by midpoint rule, trapezoidal rule, Simson’s rule with n=3 and find the exact value of integration. Express answers to five decimal places.

Want to see the full answer?

Chapter 9 Solutions

To calculate: The approximate value of integral ∫14(2x−3)3dx by midpoint rule, trapezoidal rule,

Simson’s rule with n=3 and find the exact value of integration. Express answers to five decimal places.