Drag a path through a field and watch the work accumulate. Walk it backwards and every bit of it changes sign.
Warm stretches are where the field pushes you forward; cool stretches are where you're fighting it.
Drag the path's ends; flip the direction to flip the answer.
Walk from one point to another through a field of forces. At every instant only the part of the force pointing along your direction of travel does any work — a sideways push moves you off course but costs and gains nothing. That component is F · T, and adding it up as you go is the line integral ∫ F · dr.
The graph underneath is exactly that quantity, plotted against how far along you are. Warm where the field helps, cool where it hinders, and the total is the signed area under it. Bow the path and you can watch stretches flip from one to the other.
Two things follow immediately. First, walking the path backwards negates every single contribution, so the whole integral changes sign — direction is part of the question, not an afterthought. Second, the answer depends on the path, not just the endpoints: bow it and the number moves.
Unless it doesn't. For some fields, bowing the path changes nothing at all — and that turns out to be a very special property worth chasing down.