Explanation Given information: I = ∫ 1 3 ∫ 0 1 y e x y d y d x Formula used: The integral formula by parts formula ∫ u d v d t d t = u v − ∫ u ′ v d t The integral formula ∫ e x d x = e x And ∫ e x ( 1 x − 1 x 2 ) d x = e x x + C The differentiation formula d d x ( x n ) = n x n − 1 Calculation: To calculate the integral I = ∫ 1 3 ∫ 0 1 y e x y d y d x First, evaluate inner integral treating x as constant Thus, S ( x ) = ∫ 0 1 y e x y d y , to evaluate this first Substitute x y = t x d y = d t , limit changes to, When y = 0 ; t = 0 y = 1 ; t = x Thus integral changes as, S ( x ) = ∫ 0 x t x 2 e t d y = 1 x 2 ∫ 0 x t e t d y Now to evaluate ∫ 0 x t e t d t , use integration by parts treating u = t & d v d t = e t , thus v = ∫ e t d t = e t Now according to integration by parts formula ∫ u d v d t d t = u v − ∫ u ′ v d t Thus, ∫ 0 x t e t d t = [ t e t − ∫ ( t ) ′ e t d t ] 0 x = [ t e t − ∫ e t d t ] 0 x = [ t e t − e t ] 0 x = x e x − e x + 0 + e 0 = x e x − e x + 1 Thus S ( x ) = 1 x ∫ 0 x t e t d t = 1 x 2 ( x e x − e x + 1 ) = e x x − e x x 2 + 1 x 2 = e x ( 1 x − 1 x 2 ) + 1 x 2 Now, integrate S(x) with respect to x, thus

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ISBN: 9781319055905

Author: FRANZOSA

Publisher: VST

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Question

Chapter 15.1, Problem 46E

To determine

To evaluate the iterated integral

I=∫13∫01yexydydx

And integrate this again using Fubini’s Theorem to change the order of integration(integrate with respect to x first)

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Chapter 15.1, Problem 45E

Chapter 15.1, Problem 47E

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Chapter 15 Solutions

EBK CALCULUS: EARLY TRANSCENDENTALS

Ch. 15.1 - Prob. 1PQ Ch. 15.1 - Prob. 2PQ Ch. 15.1 - Prob. 3PQ Ch. 15.1 - Prob. 4PQ Ch. 15.1 - Prob. 5PQ Ch. 15.1 - Prob. 6PQ Ch. 15.1 - Prob. 1E Ch. 15.1 - Prob. 2E Ch. 15.1 - Prob. 3E Ch. 15.1 - Prob. 4E

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To evaluate the iterated integral I = ∫ 1 3 ∫ 0 1 y e x y d y d x And integrate this again using Fubini’s Theorem to change the order of integration (integrate with respect to x first)

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To evaluate the iterated integral I=∫13∫01yexydydx And integrate this again using Fubini’s Theorem to change the order of integration(integrate with respect to x first)

Want to see the full answer?

Chapter 15 Solutions

To evaluate the iterated integral

I=∫13∫01yexydydx

And integrate this again using Fubini’s Theorem to change the order of integration(integrate with respect to x first)