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Finding limits of line integral

  1. May 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Integrate along the line segment from (0,0) to [itex](\pi,-1)[/itex]
    The integral

    [itex]\int_{(0,1)}^{(\pi,-1)} [y sin(x) dx - (cos(x))]dy[/itex]



    2. Relevant equations



    3. The attempt at a solution

    I have used the parameterization of [itex]x=\pi t [/itex] and [itex]y= 1-2t [/itex]
    To get the integral:
    [itex]\int_{(0,1)}^{(\pi,-1)} [1-2t sin(\pi t) -(cos(\pi t))]dt[/itex]

    But now because it is an integral of variable t I need to change the limits .

    I'm not sure if I just have to put the limits of t just from 0 to [itex] \pi [/itex]

    I suppose I'm having trouble with getting from the limit of 2 variables (x,y) to a limit of one variable t

    Thanks
     
  2. jcsd
  3. May 22, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If x=pi*t and y=1-2t, then if you put t=0 then x=0 and y=1, right? If you put t=1 then x=pi and y=(-1), also right? As you came up with that fine parametrization what's the problem with finding limits for t?
     
  4. May 23, 2009 #3
    Thank you for your help.
    I will need to go back and study more about parametrization.
     
  5. May 23, 2009 #4
    When calculating this line integral

    [itex]\int_{(0,1)}^{(\pi,-1)} [y sin(x) dx - (cos(x))]dy[/itex]

    I'm using the formula
    [itex]\int_{a}^{b}[f(x(t),y(t))x'(t) + g(x(t),y(t))y'(t)]dt [/itex]

    with parameterization

    I have [itex]x = \pi t [/itex]
    [itex]y = 1-2t[/itex]
    so
    [itex]x' = \pi [/itex]
    and
    [itex]y' = -2 [/itex]

    plugging into the integral I get

    [itex]\int_{(0)}^{(1)} [1-2y sin(\pi t) \pi - (cos(\pi t))-2][/itex]
    [itex] = -1[/itex]

    The question states that the integral is independant of path.

    So if I integrate along the initial line segment [itex](0,1)[/itex]to [itex](0,\pi)[/itex]
    I should be able to plug in the values f([itex](-1,\pi)[/itex])-f([itex](0,1)[/itex])
    And it should equal my original integral vaue of -1.

    However I get 0

    Could someone please check what I've done I show me where I am going wrong ?
     
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