Explanation Given: The given function is f ( x , y ) = y e − x z . The curve is C : r ( t ) = 〈 t , 3 t , − 6 t 〉 for 0 ≤ t ≤ ln ( 8 ) . Definition used: “Suppose f be continuous function on a region containing a smooth curve C : r ( t ) = 〈 x ( t ) , y ( t ) , z ( t ) 〉 , in the interval a ≤ t ≤ b . Then ∫ C f d s = ∫ a b f ( x ( t ) , y ( t ) ) | r ′ ( t ) | d t ”. Calculation: The curve r ( t ) = 〈 t , 3 t , − 6 t 〉 is defined in the interval 0 ≤ t ≤ ln ( 8 ) . Therefore, x ( t ) = t , y ( t ) = 3 t and z ( t ) = − 6 t . Compute the value of r ′ ( t ) . r ′ ( t ) = 〈 ( t ) ′ , ( 3 t ) ′ , ( − 6 t ) ′ 〉 = 〈 1 , 3 , − 6 〉 Obtain the value of | r ′ ( t ) | . | r ′ ( t ) | = x ′ ( t ) 2 + y ′ ( t ) 2 + z ′ ( t ) 2 = ( 1 ) 2 + ( 3 ) 2 + ( − 6 ) 2 = 1 + 9 + 36 = 46 Thus, the value of | r ′ ( t ) | = 46 . Obtain the function f ( x , y , z ) in terms of t . f ( x , y , z ) = y e − x z = ( 3 t ) e − ( t ) ( − 6 t ) = 3 t e 6 t 2 Obtain the value of ∫ C y e − x z d s

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 14, Problem 9RE

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Calculus: Early Transcendentals (2nd Edition)

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The line integral ∫ C y e − x z d s .

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To evaluate: The line integral ∫Cye−xz ds .

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Chapter 14 Solutions