I use an example from Lamar University they have a work function F(x) = kx and determined the spring constant k to be 400 so the work is equal to the integral of 400xdx, And the answer is 200x^2. Skipping some details clearly the answer will be an AREA as we expected and expressed in Joules. The units work out very well. I am a happy camper.
Now let use take the line integral over a region where x = t and y = t^2. And the force is expressed as F= -yi + xj. where i and j are the unit vectors of course. This will be the integral of F dot product dr. With respect to t this will clearly be the integral of F doted with drdt/dt and you can choose any value you wish to integrate over.
In what sense is this an area representing work? That is the essence of my question? How to I unify these two apparently disparate ways of expressing work? Where are the Joules in the equation F= -yi + xj?