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Homework Help: Line integral

  1. Jun 13, 2010 #1
    1. The problem statement, all variables and given/known data
    hello again, sorry for asking so many questions, i just want to make sure if im correct or not

    calculate the line integral y^2dx+x^2dy where line C is the triangle with sides x=1, y=0 and y=x

    3. The attempt at a solution

    first of all i tried to find a customization of the line

    we know that x = 1 hence it will be like this

    r(t) = (1,t) but im not sure if it's correct

    then i said that the integral would be this


    could I just use Green's theorem?

    I mean using Greens theorem I get the same result


    im 99% sure that greens theorem is correct, i mean the way i implemented it, but is the first way i showed also correct?

    thanks in advance
    Last edited by a moderator: Apr 25, 2017
  2. jcsd
  3. Jun 13, 2010 #2


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    "we know that x=1"? Did you leave out much of the problem? We know that x= 1, y= any number from 0 to 1 is one side of triangle and so one part of the path over which we want to integrate. Taking x= 1, y= t as parametric equations, dx= 0, dy= dt so the integral becomes
    [tex]\int_0^1 1 dx= 1[/itex]

    But you still have to do the other two sides of the triangle.

    On the line y= 0, we can use parametric equations x= t, y= 0 with t from 0 to 1. Then dx= dt, dy= 0 but [itex]y^2 dx= 0dt[/itex] so the integral is
    [tex]\int_0^1 0dt= 0[/itex].

    On the line y= x, where we are integrating from (1, 1) to (0, 0) (we got counterclockwise around the closed path), we can take x= t, y= t so that dx= dt, dy= dt and the integral is
    [tex]\int_1^0 2t^2 dt= -\int_0^1 2t^2 dt= -2/3[/itex] and the entire integral is 1- 2/3= 1/3.

    Last edited by a moderator: Apr 25, 2017
  4. Jun 13, 2010 #3
    thanks a lot for your help

    that cleared up everything in my mind
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