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Math Help - Finding parametrizattion for a Line Integral

  1. #1
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    Finding parametrizattion for a Line Integral

    I've been asked to evaluate the line integral

     \int (10x^4 - 2xy^3)dx - 3x^2y^2dy

    Over the curve

     x^4 - 6xy^3 - 4y^2 = 0

    Between the points (0,0) and (2,1).

    What I'm trying to do is finding a parametrization x(t) from the given curve and then put that into the integral so I can solve it.

    However I can't seem to make it work. I tried solving x for y, so I can say x=t and y is a function of t. Or the other way round would work as well. But the crossterm 6xy^3 gets in the way. And the powers of both x and y aren't right to use the ABC formula.

    Could anyone help me out on what to do here?
    Thanks in advance

    Edit: Apologies for the spelling mistake in the thread title. Can't edit it, it seems.
    Edit 2: Got one sign in the integral wrong.
    Last edited by AbAeterno; May 29th 2010 at 10:08 AM.
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  2. #2
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    Can I ask to check on the sign of one of your terms in the integral

    i.e. is it <br />
\int \limits_{c} (10x^4 {\color{red}{+}}\, 2xy^3)dx + 3x^2y^2dy
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    Ah, I did write it down incorrectly, but it's not that either. Correct version is:

    <br />
\int (10x^4 - 2xy^3)dx - 3x^2y^2dy<br />
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  4. #4
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    Quote Originally Posted by AbAeterno View Post
    Ah, I did write it down incorrectly, but it's not that either. Correct version is:

    <br />
\int (10x^4 - 2xy^3)dx - 3x^2y^2dy<br />
    Note that 2xy^3dx + 3 x^2 y^2 dy = d(x^2y^3).
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    Quote Originally Posted by Danny View Post
    Note that 2xy^3dx + 3 x^2 y^2 dy = d(x^2y^3).
    Okay. So I can write that down in the integral and get

    <br />
\int 10x^4 dx - d(x^2y^3)<br />
    But then what? Still doesnt help me find a parametrisation as far I can see. Or is that not what I'm supposed to do?

    Edit: oh wait.. so now I don't need to find a x(t) and y(t) anymore, but an x(t) and x^2y^3(t)?
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  6. #6
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    Quote Originally Posted by AbAeterno View Post
    Okay. So I can write that down in the integral and get

    <br />
\int 10x^4 dx - d(x^2y^3)<br />
    But then what? Still doesnt help me find a parametrisation as far I can see. Or is that not what I'm supposed to do?
    Since the vector field is conservative, the parameterization doesn't matter. So

    <br />
\int \limits_c 10x^4 dx - d(x^2y^3) = <br />
\int \limits_c d(2x^5 - x^2y^3) = \left. 2x^5 - x^2y^3 \right|_{(0,0)}^{(2,1)}<br />
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  7. #7
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    Ooh.. I totally didn't see that. Thanks!
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