Explanation Given information: ∫ − 1 1 e 2 x d x with n = 2 , 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 subintervals a ≤ x ≤ b and Δ x = b − a n Calculation: For n = 2 Compare ∫ a b f ( x ) d x to ∫ − 1 1 e 2 x d x ; it becomes a = − 1 and b = 1 Therefore, Δ x = b − a n = 1 − ( − 1 ) 2 = 1 Now, a 0 = a ⇒ a 0 = − 1 a 1 = a 0 + Δ x = − 1 + 1 = 0 a 2 = a 1 + Δ x = 0 + 1 = 1 By using the trapezoidal rule, ∫ − 1 1 e 2 x d x ≈ [ f ( − 1 ) + 2 f ( 0 ) + f ( 1 ) ] × ( 1 2 ) = [ e 2 ( − 1 ) + 2 e 2 ( 0 ) + e 2 ( 1 ) ] × ( 1 2 ) = [ 0.13534 + 2 + 7.38906 ] × ( 1 2 ) = 4.76219 ⇒ ∫ − 1 1 e 2 x d x ≈ 4.76219 For n = 4 Compare ∫ a b f ( x ) d x to ∫ − 1 1 e 2 x d x ;it becomes a = − 1 and b = 1 Therefore, Δ x = b − a n = 1 − ( − 1 ) 4 = 1 2 = 0

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Author: Goldstein

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Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differ…

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Chapter 9.4, Problem 14E

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To calculate: The approximate value of the integral ∫−11e2xdx by trapezoidal rule with n=2 ,n=4 and the exact value of the integral ∫−11e2xdx. Express the answer to five decimal places.

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Chapter 9.4, Problem 13E

Chapter 9.4, Problem 15E

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The approximate value of the integral ∫ − 1 1 e 2 x d x by trapezoidal rule with n = 2 , n = 4 and the exact value of the integral ∫ − 1 1 e 2 x d x . Express the answer to five decimal places.

Solution Summary: The author calculates the approximate and exact value of the integral by trapezoidal rule.

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To calculate: The approximate value of the integral ∫−11e2xdx by trapezoidal rule with n=2 ,n=4 and the exact value of the integral ∫−11e2xdx. Express the answer to five decimal places.

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Chapter 9 Solutions