# Need help with line integral

1. Feb 28, 2012

### mottov2

1. The problem statement, all variables and given/known data
Find the parameterized curve gamma and vector field F so that
the $\int$$\gamma$ F ds = $\int$$\int2xy dx dy$ by Green's Theorem.
where -2<y<2
1-sqrt(4-y2) < x < 1+sqrt(4-y2)

3. The attempt at a solution

x = 1 + sqrt(4-y2)
(x-1)2=4-y2
(x-1)2+y2=4

so the path is a circle centered at (1,0) with radius of 2.
parametrize this by setting x = 1+2Cos(t) and y = 2Sin(t)

for the vector field I got F = (xy2,2x2y)
but is there more than one possible vector field?

2. Feb 28, 2012

### LCKurtz

Yes, there are infinitely many such vector fields. For example, you can modify yours like this:$$\vec F = \langle xy^2+g(x), 2x^2y+h(y)\rangle$$where h and g can be any differentiable functions.