Finding parametrizattion for a Line Integral

May 2010
5
1
I've been asked to evaluate the line integral

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

Over the curve

\(\displaystyle x^4 - 6xy^3 - 4y^2 = 0 \)

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 \(\displaystyle 6xy^3\) 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:

Jester

MHF Helper
Dec 2008
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Can I ask to check on the sign of one of your terms in the integral

i.e. is it \(\displaystyle
\int \limits_{c} (10x^4 {\color{red}{+}}\, 2xy^3)dx + 3x^2y^2dy\)
 
May 2010
5
1
Ah, I did write it down incorrectly, but it's not that either. Correct version is:

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

Jester

MHF Helper
Dec 2008
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1,255
Conway AR
Ah, I did write it down incorrectly, but it's not that either. Correct version is:

\(\displaystyle
\int (10x^4 - 2xy^3)dx - 3x^2y^2dy
\)
Note that \(\displaystyle 2xy^3dx + 3 x^2 y^2 dy = d(x^2y^3)\).
 
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May 2010
5
1
Note that \(\displaystyle 2xy^3dx + 3 x^2 y^2 dy = d(x^2y^3)\).
Okay. So I can write that down in the integral and get

\(\displaystyle
\int 10x^4 dx - d(x^2y^3)
\)
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)?
 

Jester

MHF Helper
Dec 2008
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1,255
Conway AR
Okay. So I can write that down in the integral and get

\(\displaystyle
\int 10x^4 dx - d(x^2y^3)
\)
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

\(\displaystyle
\int \limits_c 10x^4 dx - d(x^2y^3) =
\int \limits_c d(2x^5 - x^2y^3) = \left. 2x^5 - x^2y^3 \right|_{(0,0)}^{(2,1)}
\)
 
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