Explanation 1) Concept: Use the Net Change theorem and the Midpoint Rule. 2) Theorem and Rule: The Net Change theorem: The integral of a rate of change is the net change. ∫ a b F ' ( x ) d x = F b - F ( a ) The Midpoint Rule: ∫ a b f x d x ≈ ∑ i = 1 n f x i - ∆ x = ∆ x f x 1 - + … + f x n - where ∆ x = b - a n and x i - = 1 2 x i - 1 + x i midpoint of x i - 1 , x i . 3) Calculation: Power is the rate of change of energy with respect to time, that is, P t = E ' t Here, t ∈ 0,24 By the Net Change theorem, ∫ 0 8 P t d t ≈ ∫ 0 8 E ' t d t = E 8 - E ( 0 ) is energy used on December 9 , 2004 at the time interval. Approximate the value of the integral using the Midpoint Rule with n = 12 , a = 0 , and b = 24 . Then the width of interval is ∆ t = b - a n Substitute values. ∆ t = 24 - 0 12 ∆ t = 2 From the graph, the three subintervals are 0,2 , 2,4 , 4,6 , … , 20,22 , 23,24 and the midpoints are t 1 - = 1 2 t 0 + t 2 = 1 2 0 + 2 = 1 2 2 = 1 t 2 - = 1 2 t 2 + t 4 = 1 2 2 + 4 = 1 2 6 = 3 t 3 - = 1 2 t 4 + t 6 = 1 2 4 + 6 = 1 2 10 = 5

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ISBN: 9781337051545

Author: Stewart

Publisher: CENGAGE C

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Chapter 4.4, Problem 67E

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The energy used onDecember 9, 2004.

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Chapter 4.4, Problem 66E

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To estimate:

The energy used onDecember 9, 2004.