Integrals

Adding up infinitely many pieces.

Accumulation

An integral totals a quantity that varies. Slice the interval into many thin pieces, add them, and take the limit as the slices get thinner.

Geometrically it is the area under the curve.
Distance is the integral of speed; total charge is the integral of current.

The two kinds

A definite integral gives a number over a range. An indefinite integral gives a function whose derivative is the one you started with, plus a constant.