Archimedes in the third century BCE computed the area of a parabolic segment by exhausting it with inscribed triangles, summing their areas, and arguing that the sum approached the true area as the triangles got smaller. He was, in effect, doing integration without the notation. Two thousand years later, Newton and Leibniz completed the picture: integration and differentiation are inverse operations, joined by what is now called the Fundamental Theorem of Calculus. The discovery was so consequential that it gave its inventors a priority dispute lasting a century. The result, however arrived at, is the second of the two engines of modern mathematics — the one that totals what the derivative differentiates.
There are two complementary ways to read the integral. The Riemann integral defines ∫ₐᵇ ƒ(x) dx as the limit of sums: partition [a,b] into many small subintervals, sum the products of (subinterval width) × (function value somewhere inside), and let the partition get arbitrarily fine. If the limit exists and doesn't depend on how the function values are chosen, the function is Riemann integrable and the limit is its definite integral — a single number, the signed area between the curve and the x-axis. The antiderivative reads the integral the other way: an indefinite integral ∫ ƒ(x) dx is any function whose derivative is ƒ. The Fundamental Theorem of Calculus unifies these: ∫ₐᵇ ƒ(x) dx = F(b) − F(a), where F is any antiderivative of ƒ. The theorem is almost magical — it says that to compute an area built from infinitely many infinitesimal pieces, you need only evaluate a related function at two points. Beyond Riemann lies Lebesgue integration (1902), which handles wildly more pathological functions by reorganizing the sum: instead of partitioning the domain, partition the range and ask how much of the domain maps to each range bin. Lebesgue integration generalizes to measure theory, the foundation of modern probability (the integral over a probability measure is the expectation of a random variable). Properties of the integral — linearity, monotonicity, the change-of-variables formula — are the working calculus of every applied science.
Integration is everywhere computation needs a total of a varying quantity. Energy in physics is the integral of force over distance. Probability is an integral of a density. Expected return in finance is an integral over outcome distributions. Image rendering in graphics integrates light transport equations. Numerical quadrature — Simpson's rule, Gauss-Legendre, Monte Carlo integration — is one of the most-used algorithms in scientific computing. The Lebesgue framework underwrites the entire theory of stochastic processes that runs modern finance, signal processing, and machine learning's theoretical foundations.