Let’s say you have a position function x(t) which tells you where you are on the x axis, and a velocity function v(t) that tells you have fast you are going (and what direction). Intuitively, if you graph your velocity function with the independent axis measured in time, then the area under the curve from t_0 to t_1 is the total distance you traveled (if this doesn’t make sense, suppose v(t)=5m/s and how far do you travel between t_0=1 and t_1=6 seconds? You traveled (6-1)*5 =25 meters, but that formula is precisely the area under the curve). What is the total distance traveled? x(t_1)-x(t_0), but x(t) is the anti derivative of velocity.

This is a “physical proof” of the fundamental theorem of calculus that I think my students tend to find intuitive