A function built by stacking area — its rate of change is the original function
Turn the area accumulated up to x into a function, and that area function's instantaneous rate of change turns out to be exactly the same as the original function f(x). See with your own eyes that differentiation and integration are opposite processes.
The area function A(x) = ∫₀ˣ f(t)dt is "the area accumulated from 0 up to x." Move x with the slider, watch the area pile up on the graph above (f), and see how that value gets traced out on the graph below (A). The slope of A(x)'s tangent line is always equal to f(x) — that's the Fundamental Theorem of Calculus.
x1.0
f(t) = t, A(x) = ∫₀ˣ t dt
An antiderivative refers to any function that becomes f(x) after differentiating. The antiderivative of f(x)=2x is F(x)=x²+C — no matter what value C takes, differentiating makes the constant vanish, so you always get 2x back. That's why an antiderivative isn't a single function, but an entire family of curves shifted up and down from each other.