An integral is a mathematical construction that combines local contributions into an accumulated quantity. In calculus, a definite integral assigns a value to a function over an interval, often interpreted as signed area or total change. An indefinite integral describes a family of functions whose derivative equals a given function. These meanings are connected by the fundamental theorem of calculus, but integration can be defined independently of antidifferentiation. (openstax.org)
Definite integrals and accumulation
For a bounded, real-valued function (f) on ([a,b]), divide the interval into subintervals with endpoints [ a=x_0<x_1<\cdots<x_n=b. ] Choose a sample point (\xi_i) in each subinterval and form [ \sum_{i=1}^{n}f(\xi_i)(x_i-x_{i-1}). ] This Riemann sum approximates accumulation by adding rectangular contributions. The Riemann integral [ \int_a^b f(x),dx ] exists when these sums approach the same finite limit as the largest subinterval width tends to zero, regardless of the partitions and sample points. Here (f) is the integrand, (a,b) are the limits of integration, and (x) is the integration variable. (openstax.org)
When (f\geq0), the integral represents the area beneath its graph. Negative values contribute negatively, so the integral generally measures signed rather than total geometric area. Every continuous function on a closed, bounded interval is Riemann integrable; continuity is sufficient but not necessary. (openstax.org)
Integration is linear: [ \int_a^b(\alpha f+\beta g),dx =\alpha\int_a^b f,dx+\beta\int_a^b g,dx. ] It is also additive across adjacent intervals, and reversing the limits changes its sign: [ \int_a^b f,dx=\int_a^c f,dx+\int_c^b f,dx, \qquad \int_b^a f,dx=-\int_a^b f,dx. ] These properties apply whenever the relevant integrals exist. (openstax.org)
Antiderivatives and the fundamental theorem
An antiderivative of (f) on an interval is a function (F) satisfying (F'=f). Any two antiderivatives on that interval differ by a constant, giving the notation [ \int f(x),dx=F(x)+C. ] Unlike a definite integral with fixed limits, this expression denotes a family of functions, not a single number. (openstax.org)
The fundamental theorem of calculus establishes the connection. If (f) is continuous on ([a,b]), the accumulation function [ A(x)=\int_a^x f(t),dt ] satisfies (A'(x)=f(x)) for interior points. Moreover, for any antiderivative (F), [ \int_a^b f(x),dx=F(b)-F(a). ] Thus, a quantity defined through limiting sums can be evaluated using differentiation in reverse. For example, [ \int_0^1 x^2,dx =\left[\frac{x^3}{3}\right]_0^1 =\frac13. ] (openstax.org)
Evaluation and approximation
Symbolic integration uses identities derived from differentiation. Integration by substitution reverses the chain rule, while integration by parts rearranges the derivative rule for a product: [ \int_a^b u(x)v'(x),dx =[u(x)v(x)]_a^b-\int_a^b u'(x)v(x),dx. ] A suitable choice of factors can replace a difficult integral with a simpler one. (openstax.org)
Many integrals cannot be evaluated conveniently through elementary antiderivatives. Numerical integration, also called quadrature, estimates definite integrals from sampled values. Midpoint and trapezoidal rules use constant or linear approximations on subintervals; Simpson’s rule uses quadratic approximations. Under appropriate smoothness assumptions, error bounds relate accuracy to subinterval width and bounds on derivatives. Numerical approximation is distinct from determining whether an integral exists. (openstax.org)
Improper and Lebesgue integrals
An improper integral extends integration to unbounded intervals or unbounded integrands through limits. For example, [ \int_1^\infty f(x),dx =\lim_{R\to\infty}\int_1^R f(x),dx, ] provided the limit is finite. The integral of (x^{-2}) over this interval converges to (1), whereas that of (x^{-1}) diverges. Singularities inside an interval require separate one-sided limits; cancellation across a singularity does not establish ordinary improper convergence. (openstax.org)
The Lebesgue integral, developed within measure theory, uses measures of sets to define accumulation. It extends integration to functions and spaces beyond the usual Riemann framework. On a bounded interval, every Riemann-integrable function is Lebesgue integrable with the same value. The function equal to (1) at rational numbers and (0) elsewhere on ([0,1]) is not Riemann integrable, but its Lebesgue integral is (0), because the rational numbers have measure zero. (math.mit.edu)
Higher dimensions and applications
Multiple integrals accumulate quantities over regions in two or more dimensions, allowing calculations of area, volume, and mass from density. A line integral accumulates quantities along a curve; integrating a force field along a path gives the work performed during motion. (openstax.org)
In probability, a probability density function (p) determines interval probabilities through [ P(a\leq X\leq b)=\int_a^b p(x),dx. ] For a random variable with a density and finite absolute expectation, its expected value is [ E[X]=\int_{-\infty}^{\infty}x,p(x),dx. ] Integration therefore provides both probability totals and weighted averages. (live.ocw.mit.edu)
Historical development
Early precursors arose in ancient Greece, where Archimedes used the method of exhaustion to establish areas and volumes through geometric approximations. In the seventeenth century, Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus, linking accumulation with rates of change. Their approaches differed, but both made the inverse relationship between integration and differentiation central. (mathshistory.st-andrews.ac.uk)
Leibniz introduced the symbol (\int) in a manuscript dated October 29, 1675. Its elongated letter S represented summa, the Latin word for “sum.” He subsequently combined it with differential notation, producing the recognizable form (\int f(x),dx). (mathshistory.st-andrews.ac.uk)