Explanation Given Info: ∫ c 2 x y z d x + x 2 z d y + x 2 y d z Over the path r ( t ) = ( t 2 , Sin ( π t 4 ) , e t 2 − 2 t ) for 0 ≤ t ≤ 2 Formula used: Integral can be converted inform of path integral by following way. ∫ c 2 x y z d x + x 2 z d y + x 2 y d z = ∫ c ( a , b , c ) ( d x , d y , d z ) ∫ c x n d x = x n + 1 n + 1 + c d d t ( u v w ) = v w d d t u + u w d d t v + u v d w d t ∫ d d t [ f ( t ) ] d t = f ( t ) Calculation: The path is as follows: r ( t ) = ( t 2 , Sin ( π t 4 ) , e t 2 − 2 t ) 0 ≤ t ≤ 2 The integral can be written in form of path in tegral, ∫ c 2 x y z d x + x 2 z d y + x 2 y d z = ∫ c ( 2 x y z d x + x 2 z d y + x 2 y d z ) ( d x , d y , d z ) Substitute x with t 2 , y with Sin ( π t 4 ) , z with e t 2 − t 2 ∫ c ( 2 x y z , x 2 z , x 2 y ) ( d x , d y , d z ) = ∫ c ( 2 t 2 sin ( π t 4 ) e t 2 − 2 t , t 4 − e t 2 − 2 t , t 4 Sin ( π t 4 ) ) ( d x , d y , d z ) = ∫ c 2 t 2 Sin ( π t 4 ) e t 2 − 2 t d x + t 4 e t 2 − 2 t d y + t 4 sin ( π t 4 ) d z Since, x = t 2 d x d t = 2 t [ differentiate w .r .t ] d x = 2 t d t y = Sin ( π t 4 ) d y d t = π 4 Cos ( π t 4 ) d y = π 4 Cos ( π 4 t ) d t z = e t 2 − 2 t d z d t = e t 2 − 2 t × ( 2 t − 2 ) d z d t = ( 2 t − 2 ) e t 2 − 2 t d z = ( 2 t − 2 ) e t 2 − 2 t d t Substitute the value of d x 1 , d y 1 d z to get the function in one form = ∫ c 2 t 2 Sin ( π t 4 ) ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 17.3, Problem 19E

To determine

The value of the integral $\int_{c} 2 x y z d x + x^{2} z d y + x^{2} y d z$ over the path

$r (t) = (t^{2}, Sin (\frac{π t}{4}), e^{t^{2} - 2 t}) for 0 \leq t \leq 2$

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Chapter 17.3, Problem 18E

Chapter 17.3, Problem 20E

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Chapter 17 Solutions

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The value of the integral ∫ c 2 x y z d x + x 2 z d y + x 2 y d z over the path r ( t ) = ( t 2 , Sin ( π t 4 ) , e t 2 − 2 t ) for 0 ≤ t ≤ 2

Solution Summary: The author explains that the integral can be written in form of path in t.

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The value of the integral $\int_{c} 2 x y z d x + x^{2} z d y + x^{2} y d z$ over the path

$r (t) = (t^{2}, Sin (\frac{π t}{4}), e^{t^{2} - 2 t}) for 0 \leq t \leq 2$

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Chapter 17 Solutions

The value of the integral ∫ c 2 x y z d x + x 2 z d y + x 2 y d z over the path r ( t ) = ( t 2 , Sin ( π t 4 ) , e t 2 − 2 t ) for 0 ≤ t ≤ 2

Solution Summary: The author explains that the integral can be written in form of path in t.

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The value of the integral ∫c2xyzdx+x2zdy+x2ydz over the path r(t)=(t2,Sin(πt4),et2−2t) for 0≤t≤2

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Chapter 17 Solutions

The value of the integral $\int_{c} 2 x y z d x + x^{2} z d y + x^{2} y d z$ over the path

$r (t) = (t^{2}, Sin (\frac{π t}{4}), e^{t^{2} - 2 t}) for 0 \leq t \leq 2$