Integration is the reverse of differentiation — and also the way to compute area under a curve.
Drag the right edge
y = 0.4·x·x + 1area [0, 2.5] ≈ 4.58
Two faces of one idea
Integration answers two questions at once: what function has this derivative? (the antiderivative) and how much area sits under this curve? Drag the edge of the shaded region above — the number it reports is the running total of area, and that number *is* the integral.
Indefinite integral
F is any antiderivative of f; the +C covers every vertical shift, since constants vanish under differentiation.
The ∫ sign is a stretched-out letter S — for *sum*. An integral is the limit of summing up infinitely many infinitely thin slices.
Your turn
∫ x dx = ?
These two ideas (antiderivative and area) are linked by the Fundamental Theorem of Calculus.