ok....integration of a vector F field along a curve...(exam fall-out)

means that you have to integrate the F along the curve.  That means that you need to find a parameterization of the curve. Usually there will be one of three situations: 1) we will give you the parameterization--use it, or 2) there is a natural parameterization of the curve, i.e. the curve is a circle or the curve is the graph of a function g(x)--use <cos(t),sin(t)> for the circle or use <t,g(t)> for the function, or 3) the curve doesn't matter anyway because the vector field is conservative--in which case you need to find the function f(x,y) so that F=∇f and evaluate f at the endpoints of the curve.

You don't get to chose some random path that doesn't fit the curve, and don't need a path at all in the case that F is conservative.