What does the line integral of a conservative vector field really mean?

I understand that a line integral of a vector function is the area under the curve, but I'm confused about taking the line integral of a conservative vector field because it's independent of the curve. So what does the "answer" really "mean"?|||A physical interpretation is work. You compute the work done moving a particle from one end of the path to the other subject to the force given by the vector field. For a conservative field, the work is independent of the path chosen. But the interpretation of work is equally valid for conservative and non-conservative force fields (in the non-conservative case, you have to integrate along the prescribed path--different paths can lead to different work done.)