Explanation Given information: ∫ 0 10 e x 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 the midpoint rule and T is the approximate value of integral calculated by Trapezoidal rule. Calculation: Compare ∫ a b f ( x ) d x with ∫ 0 10 e x d x , it becomes a = 0 and b = 10 . Therefore, Δ x = b − a n = 10 − 0 5 = 2 . The endpoints of the intervals begin at a = 0 and are spaced 2 units apart. The first midpoint is at a + Δ x 2 = 0 + 2 2 = 1 ⇒ x 1 = 1 The midpoints are also spaced 2 units apart. Therefore x 2 = 3 , x 3 = 5 , x 4 = 7 and x 5 = 9 . By using the midpoint rule, the integral is approximately equal to ∫ 0 10 e x d x ≈ [ e 1 + e 3 + e 5 + e 7 + e 9 ] ( 2 ) = 2 [ 2

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Chapter 9, Problem 44RE

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To calculate: The approximate value of the integral ∫010exdx 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 43RE

Chapter 9, Problem 45RE

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The approximate value of the integral ∫ 0 10 e x d x by midpoint rule, trapezoidal rule, Simpson’s rule with n = 5 where n is the number of equal subintervals.

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To calculate: The approximate value of the integral ∫010exdx 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 Solutions