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Line integral with respect to x or y

  1. Aug 27, 2007 #1
    I am confused about how

    [tex] \int_C f(x,y) dx = \lim_{||P|| \to 0} \sum_{i = 1}^n f(x_i^*,y_j^*) \Delta x_i [/tex] is different from [tex] \int f(x,y) dx [/tex]

    where P is a partition and its norm is the length of its largest elements. The index i represents an element in that partition and the asterik means the endpoint closest to the origin of that part of the partition.
     
  2. jcsd
  3. Aug 27, 2007 #2

    quasar987

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    Is there a context? Where did you see that they were different things? The classical notation for a line integral in the plane is

    [tex]\int_C P(x,y)dx+Q(x,y)dy[/tex]

    where this is to be understood as

    [tex]\int_{a}^{b}P(x(t),y(t))\frac{dx}{dt}dt+\int_{a}^{b}Q(x(t),y(t))\frac{dy}{dt}dt[/tex]

    Where t is the parameter for the curve C.

    But if there is no Q(x,y), the integral turns out to be equivalent to a simple integration over x like you wrote.
     
  4. Aug 28, 2007 #3
    So, you are saying

    [tex]\int_{a}^{b}P(x(t),y(t))\frac{dx}{dt}dt = \int_{x(a)}^{x(b)}P(x,y)dx [/tex]

    That seems unintuitive to me for some reason.
     
  5. Aug 28, 2007 #4

    quasar987

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    It's just the change of variable formula!
     
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