Mathematics
Line Integrals and Conservative Vector Fields
Quick fact
Line integrals of a conservative vector field depend only on the endpoints of the curve, not the path taken. This means if you're moving a particle through a gravitational or electric field, the work done is simply the difference in potential energy between the start and end points—regardless of the route.
Why this is interesting
You've calculated the area under a curve, but what if the curve isn't the x-axis and the value being accumulated changes direction at every point? That's when line integrals come in—and they hold a surprising secret that can turn a painstaking calculation into a simple subtraction.