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Homework Help: Integrals 3

  1. Jan 25, 2012 #1
    1. The problem statement, all variables and given/known data

    Evaluate this integral along the line segment from (0,1) to (pi, -1) by parameterising this segment

    2. Relevant equations

    [itex]\int y sin x dx - cos x dy[/itex]

    3. The attempt at a solution

    How would I parameterise this line segment?
     
  2. jcsd
  3. Jan 25, 2012 #2

    SammyS

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    Start by writing an equation for the line passing through those two points.
     
  4. Jan 25, 2012 #3
    [itex]y= \frac{-2x}{\pi} +1[/itex]. How do I know whether to let x=t or not?
     
  5. Jan 25, 2012 #4

    tiny-tim

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    hi bugatti79! :smile:
    you can let t = x, or t = y, or t = any function of x and y so long as it's strictly increasing (or decreasing) along the line :wink:
     
  6. Jan 25, 2012 #5

    SammyS

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    You may want t = 0 to correspond to x = -1 (the left end of the interval) and t = 1 to correspond to x = 1 (the right end of the interval).

    That's a fairly common practice.
     
  7. Jan 26, 2012 #6
    Very good, thats a good tip. Will keep that in mind.

    I dont follow this to be honest. I am not sure what the line is. Is it an arc from (0,1) to (pi, -1)?

    Does this mean the integral will evaluate to 0? I think that is only for closed simple curves and we are dealing with an open line hence it will be none 0..right?

    What is the significant of whether it was a segment or not?
     
  8. Jan 26, 2012 #7

    SammyS

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    The problem states LINE SEGMENT from (0,1) to (π, -1). That means the portion of the (straight) line passing through the points which is between (0,1) and (π, -1) including the endpoints.

    The simplest parametrization is to let x(t) = t, and [itex]\displaystyle y(t)=\frac{-2x(t)}{\pi} +1=\frac{-2t}{\pi} +1\,,[/itex] where t goes from 0 to π.

    A parametrization I like is for t to go from 0 to 1. Then x(t) = π t, and [itex]\displaystyle y(t)=\frac{-2\pi t}{\pi} +1=-2t+1\,.[/itex]
     
  9. Jan 26, 2012 #8

    Ok. I will work out both. Trying the first I get

    [itex] \displaystyle \int y sin x dx - cos x dy = \int (\frac{-2 t}{\pi}+1) sin t dt+ \frac{2}{\pi} cos t dt=0[/itex]......
     
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