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 ) ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Concept explainers

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

Videos

Question

Chapter 17.3, Problem 17E

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

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 17.3, Problem 16E

Chapter 17.3, Problem 18E

Students have asked these similar questions

48. f(x) = { 4 x if x < 2 2x 2 if x 2

Г 49. -x+1 if x 1 Answer ->

A Content X MindTap - Cengage Learning x Function Evaluations x + /ui/evo/index.html?elSBN=9780357038406&id=339416021&snapshotld=877369& GE MINDTAP , Limits, and the Derivative ⭑ វា a ANSWEI 16. Refer to the graph of the function f in the following figure. कर्ट AA C 54 -3-2 7 7 Ay 6. S 5. y=f(x) 4 3. 2. 1 -3- 34567 8 00 9 10 a. Find the value of ƒ (7). b. Find the values of x corresponding to the point(s) on the graph of ƒ located at a height of 5 units from the x-axis. c. Find the point on the x-axis at which the graph of ƒ crosses it. What is the value of f (x) at this point? d. Find the domain and range of f. MacBook Pro G Search or type URL + > % Λ & 5 6 7 29 ( 8 9 0

Chapter 17 Solutions

Calculus - Standalone book

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

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

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.

Concept explainers

Videos

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

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

Want to see the full answer?

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.

Concept explainers

Videos

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

Want to see the full answer?

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

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