Explanation Given: c = c 1 + c 2 + c 3 where c 1 is line segment AB from (2,0,0) to (0,4,0), c 2 is line segment BC from (0,4,0) to (0,0,6), c 3 is line segment CA from (0,0,6) to (2,0,0), Let I i = ∫ c i F . d r , i = 1 , 2 , 3 Formula used: Parametric equation for line segment from ( x 0 , y 0 , z 0 ) to ( x 1 , y 1 , z 1 ) is x = ( 1 − t ) x 0 + t x 1 y = ( 1 − t ) y 0 + t y 1 z = ( 1 − t ) z 0 + t z 1 for 0 ≤ t ≤ 1 d d t t n = n t n − 1 ∫ t n d t = t n + 1 n + 1 ∫ e a x d x = ( e a x / a ) + c ∫ u v d x = u ∫ v d x − ∫ d u d x ∫ v d x d x Calculation: For c 1 , using parametric equations, we get r 1 ( t ) = ( 2 − t , 4 t , 0 ) ∴ I 1 = ∫ c 1 F . d r 1 = ∫ 0 1 F ( r 1 ( t ) ) . r 1 ( t ) . d t = ∫ 0 1 ( 2 − 2 t + 4 t e 2 − 6 t ) d t = [ 2 t − t 2 − 4 t e 2 − 6 t 6 + 4 ∫ e 2 − 6 t 6 d t ] 0 1 = [ 2 t − t 2 − 4 t e 2 − 6 t 6 − 4 e 2 − 6 t 36 ] 0 1 = e 2 9 + 1 − 7 e − 4 9 For c 2 , using parametric equations, we get r 2 ( t ) = ( 0 , t , 1 ) ∴ I 2 = ∫ c 2 F ...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17.2, Problem 38E

To determine

To evaluate: The line integral of $F (x, y, z) = 〈 \begin{matrix} e^{z}, & e^{x - y}, & e^{y} \end{matrix} 〉$ over the given path

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Chapter 17.2, Problem 37E

Chapter 17.2, Problem 39E

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Chapter 17 Solutions

CALCULUS (CLOTH)

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

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The line integral of F ( x , y , z ) = 〈 e z , e x − y , e y 〉 over the given path

Solution Summary: The author evaluates the line integral of F (x, y, and z) over the given path using parametric equations.

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To evaluate: The line integral of $F (x, y, z) = 〈 \begin{matrix} e^{z}, & e^{x - y}, & e^{y} \end{matrix} 〉$ over the given path

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Chapter 17 Solutions

The line integral of F ( x , y , z ) = 〈 e z , e x − y , e y 〉 over the given path

Solution Summary: The author evaluates the line integral of F (x, y, and z) over the given path using parametric equations.

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To evaluate: The line integral of F(x,y,z)=〈ez,ex−y,ey〉 over the given path

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Chapter 17 Solutions

To evaluate: The line integral of $F (x, y, z) = 〈 \begin{matrix} e^{z}, & e^{x - y}, & e^{y} \end{matrix} 〉$ over the given path