Explanation Given: During a surge in the demand for electricity, the rate, r , at which energy is used can be approximated by r = t e − a t , where t is the time in hours and a is a positive constant. Formula Used: Integration by Parts: ∫ u v ′ d x = u v − ∫ u ′ v d x Calculation: Given that the rate, r , at which energy is used can be approximated by r = t e − a t , where t is the time in hours and a is a positive constant. Now, total energy used in first T hours is calculated as ∫ 0 T r d t = ∫ 0 T t e − a t d t Let − a t = u , − a d t = d u , d t = d u − a When t = 0 , u = 0 When t = T , u = − a T Thus, we get ∫ 0 T t e − a t d t = ∫ 0 − a T u − a × e u × d u − a = ∫ 0 − a T 1 a 2 × ( u × e u ) d u = 1 a 2 ∫ 0 − a T ( u × e u ) d u = 1 a 2 ∫ − a T 0 − ( u × e u ) d u To determine (b) To find what happens to total energy, E when T → ∞ .

6th Edition

ISBN: 9781119399353

Author: DA

Publisher: JOHN WILEY+SONS,INC.-CONSIGNMENT

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Question

Chapter 6.7, Problem 29P

To determine

(a)

To find the total energy, E, used in the first T hours.

To determine

(b)

To find what happens to total energy, E when T→∞.

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Chapter 6.7, Problem 28P

Chapter 6, Problem 1SYU

Chapter 6 Solutions

EBK APPLIED CALCULUS, ENHANCED ETEXT

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(a) To find the total energy, E , used in the first T hours.