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Homework Help: Line Integral

  1. Feb 27, 2010 #1
    1. The problem statement, all variables and given/known data

    So I'm trying to find the integral of P = 2xyz^2 along the curve c which is defined by:

    x=t^2
    y=2t
    z=t^3

    t goes from 0 to 1

    So the q says that it is the integral of P dr along c


    2. Relevant equations



    3. The attempt at a solution

    So I know that this should equal the integral of P times mod r'(t) dt..

    But if r = (t^2, 2t, t^3) then mod of r'(t) is pretty ugly..and I can't solve the integral..

    Any pointers?
     
  2. jcsd
  3. Feb 27, 2010 #2
    Line Integral of Scalar Function

    1. The problem statement, all variables and given/known data

    So I'm trying to find the integral of P = 2xyz^2 along the curve c which is defined by:

    x=t^2
    y=2t
    z=t^3

    t goes from 0 to 1

    So the q says that it is the integral of P dr along c


    2. Relevant equations



    3. The attempt at a solution

    So I know that this should equal the integral of P times mod r'(t) dt..

    But if r = (t^2, 2t, t^3) then mod of r'(t) is pretty ugly..and I can't solve the integral..

    Any pointers?
     
  4. Feb 27, 2010 #3
    Write P in terms of t?
     
  5. Feb 27, 2010 #4

    ideasrule

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    Homework Helper

    I have to admit I don't know either, but I'm fairly confident the teacher doesn't expect you to solve the integral using the "standard" method. Here's how complicated the improper integral is:

    http://integrals.wolfram.com/index.jsp?expr=x^9*sqrt%289*x^4%2B4x^2%2B4%29&random=false
     
  6. Feb 27, 2010 #5

    gabbagabbahey

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    Gold Member

    Re: Line Integral of Scalar Function

    You might want to start by completing the square on [itex]||\textbf{r}'(t)||^2[/tex] and then make an appropriate substitution. Even then, it looks like you will need to use integration by parts several times.
     
  7. Feb 27, 2010 #6

    berkeman

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    Staff: Mentor

    (Two threads merged and moved to Calculus & Beyond)

    Please do not multiple post questions here, Joe. Thanks.
     
  8. Feb 28, 2010 #7
    Hmm yes, complicated indeed!

    Is there a theorem I can use? Greens? Stokes?
     
  9. Feb 28, 2010 #8

    gabbagabbahey

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    Gold Member

    I don't see any tricks to make this one easier. Your curve isn't closed, so it doesn't bound a surface and certainly doesn't enclose a volume so Green's and Stokes' theorems won't help. The fundamental theorem for gradients also doesn't seem to help here (I cant think of a scalar function for which [itex]\mathbf{\nabla}\Phi\cdot d\textbf{r}=Pdr[/itex]...can you?) I also don't see any tricks involving the limits of integration, so it seems to me like you will have to do it the "standard way" if you are required to calculate the integral analytically.

    For the "standard way", I stand by my earlier suggestion.
     
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