Integral

Integralis in medieval Latin means 'making up a whole,' from integer (whole, complete, untouched), from in- (not) + tangere (to touch). An integer is something that has not been touched — not broken, not divided. The mathematical integral, named by Johann Bernoulli in 1690 and published by Leibniz in 1686 using the elongated S symbol (∫), is the process of summing infinitely small pieces to reconstruct a whole. The name is exact: integration is reassembly.

The integral sign ∫ is a stylized S, standing for Latin summa (sum). Leibniz introduced it in a manuscript dated October 29, 1675. The choice was deliberate: integration is summation of infinitely many infinitely small quantities. The notation compressed an entire mathematical operation into a single symbol. Newton's competing notation — dots above variables — was less expressive and eventually lost.

Integration and differentiation are inverse operations — the Fundamental Theorem of Calculus, formalized by Newton and Leibniz independently, states that integration undoes differentiation and vice versa. If differentiation breaks a function into its rates of change (its fragments), integration reassembles the fragments into the original function. The Latin etymology anticipated the mathematical relationship: fractions break apart, integrals make whole.

Integrals are everywhere in applied science. The area under a curve, the volume of a solid, the total distance traveled, the accumulated dose of radiation, the present value of a cash flow — all are integrals. The word integer also gave English 'integrity' (moral wholeness) and 'integrate' (to combine into a whole). The family of words insists on the same idea: untouched, unbroken, complete.