Integration

Integration is a core concept of calculus that formalizes accumulation. It serves as the inverse operation to differentiation and can represent quantities such as the area under a curve, the total distance from a velocity function, or any accumulated amount over an interval. Indefinite integrals yield families of antiderivatives, while definite integrals produce a number for a given interval.

The most common definition is the Riemann integral, built from limits of sums. Lebesgue integration extends this framework to a broader class of functions. Improper integrals generalize to unbounded domains or unbounded integrands and require convergence tests.

The Fundamental Theorem of Calculus ties differentiation and integration: if f is continuous on [a,b], then F(x) = ∫_a^x f(t) dt is an antiderivative of f, and ∫_a^b f(x) dx = F(b) − F(a).

Numerical methods approximate integrals when closed-form antiderivatives do not exist. Common methods include the trapezoidal rule, Simpson's rule, and Gaussian quadrature. In multiple dimensions, iterated integrals and coordinate changes (polar, cylindrical, spherical) enable volume and mass calculations.

Integration has applications across science and engineering, from geometry and physics to probability and economics. It emerged in the 17th century with Newton and Leibniz and was later formalized by Riemann and Lebesgue.