Explanation Given information: ∫ 0 4 ( x 2 + 5 ) d x with n = 2 , 4 where n is the number of equal subintervals. Formula used: Midpoint Rule: ∫ 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 Calculation: For n = 2 Compare ∫ a b f ( x ) d x to ∫ 0 4 ( x 2 + 5 ) d x ; it becomes a = 0 and b = 4 Therefore, Δ x = b − a n = 4 − 0 2 = 2 . Now, a 0 = a 1 = 0 x 1 = a 0 + Δ x 2 = 0 + 2 2 = 1 x 2 = x 1 + Δ x = 1 + 2 = 3 By using the mid-point rule, ∫ 0 4 ( x 2 + 5 ) d x = [ f ( 1 ) + f ( 3 ) ] × 2 Here f ( x ) = x 2 + 5 At x = 1 ⇒ f ( 1 ) = 1 2 + 5 = 6 At x = 3 ⇒ f ( 3 ) = 3 2 + 5 = 14 Substituting all the values in the formula, it becomes ∫ 0 4 ( x 2 + 5 ) d x = [ 6 + 14 ] × 2 = 40 For n = 4 Compare ∫ a b f ( x ) d x to ∫ 0 4 ( x 2 + 5 ) d x ; it becomes a = 0 and b = 4 Therefore Δ x = b − a n = 4 − 0 4 = 1 . Now, a 0 = a 1 = 0 x 1 = a 0 + Δ x 2 = 0 + 1 2 = 0.5 x 2 = x 1 + Δ x = 0.5 + 1 = 1.5 x 3 = x 2 + Δ x = 1

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Chapter 9.4, Problem 7E

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To calculate: The approximate value of the integral ∫04(x2+5)dx by midpoint rule with n=2 , 4 and exact value of integral ∫04(x2+5)dx. Express the answer to five decimal places.

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Chapter 9.4, Problem 6E

Chapter 9.4, Problem 8E

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The approximate value of the integral ∫ 0 4 ( x 2 + 5 ) d x by midpoint rule with n = 2 , 4 and exact value of integral ∫ 0 4 ( x 2 + 5 ) d x . Express the answer to five decimal places.

Solution Summary: The author calculates the approximate value of integral for n=2 by midpoint rule.

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To calculate: The approximate value of the integral ∫04(x2+5)dx by midpoint rule with n=2 , 4 and exact value of integral ∫04(x2+5)dx. Express the answer to five decimal places.

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Chapter 9 Solutions