Integral
Definition
The reverse of a derivative — it adds up tiny pieces to find a total. The integral of velocity over time gives the distance traveled; it is the area under the curve.
Symbol: ·
Technical definition
The definite integral from a to b of f(x) is the limit of a sum of f(x)·Δx over the interval — geometrically the signed area under the curve. By the fundamental theorem of calculus, integration inverts differentiation.
Example
A velocity-time graph shaped as a triangle with base 10 s and height 20 m/s encloses area ½ × 10 × 20 = 100 m — so the object traveled 100 meters, found by integration without any formula for position.
Integration is not just 'area' in the casual sense — it is signed area, so parts below the axis subtract. And while differentiation gives a rate, integration gives an accumulation; students who memorize 'integral = area' sometimes forget it also undoes a derivative, which is how you recover position from velocity.