Integral

An integral is a mathematical object that represents accumulation. In calculus, integration associates a function with quantities such as signed area, total change, mass, probability, or work. The concept encompasses the definite integral, which produces a number over a specified domain, and the indefinite integral, which denotes a family of functions whose derivatives equal a given function.

For a real-valued function (f) on an interval ([a,b]), the definite integral is conventionally written

[ \int_a^b f(x),dx. ]

The elongated symbol (\int) derives from a stylized letter (S), reflecting the interpretation of integration as continuous summation. The expression (f(x)) is the integrand, (a) and (b) are the limits of integration, and (dx) identifies the variable with respect to which accumulation is measured. Although this notation resembles multiplication, (dx) acquires its precise meaning from the particular theory of integration in use.

Geometric and accumulative interpretation

When (f) is nonnegative and sufficiently regular, (\int_a^b f(x),dx) equals the area bounded by the graph (y=f(x)), the (x)-axis, and the vertical lines (x=a) and (x=b). For a function that assumes negative values, the integral represents signed area: regions below the axis contribute negatively rather than being converted to positive geometric area.

The broader interpretation concerns accumulated density. If (\rho(x)) is the linear mass density of an object occupying ([a,b]), then its total mass is

[ M=\int_a^b \rho(x),dx. ]

Here the geometric appearance of the graph is secondary. The integral combines local contributions (\rho(x),dx) across the interval, with the limiting process removing dependence on any finite subdivision.

This interpretation also accounts for orientation. Reversing the endpoints changes the sign:

[ \int_b^a f(x),dx=-\int_a^b f(x),dx, ]

while an interval of zero length contributes no accumulation:

[ \int_a^a f(x),dx=0. ]

Additivity over adjacent intervals gives

[ \int_a^c f(x),dx

\int_a^b f(x),dx+\int_b^c f(x),dx. ]

These properties characterize integration as an additive assignment to oriented domains.

Riemann integration

The Riemann integral formalizes integration through finite sums. A partition

[ a=x_0<x_1<\cdots <x_n=b ]

divides ([a,b]) into subintervals. Selecting a sample point (x_i^*) in each subinterval produces the Riemann sum

[ \sum_{i=1}^{n} f(x_i^*)\Delta x_i, \qquad \Delta x_i=x_i-x_{i-1}. ]

Each term combines a sampled function value with the width of the corresponding subinterval. The function is Riemann integrable when these sums converge to a common value as the maximum subinterval width approaches zero, independently of the permitted sample-point choices.

Every continuous function on a closed bounded interval is Riemann integrable. Bounded functions with sufficiently limited discontinuity are also integrable; more precisely, the Lebesgue criterion for Riemann integrability states that a bounded function is Riemann integrable exactly when its set of discontinuities has measure zero.

The Riemann construction preserves linearity. For integrable functions (f) and (g), and real constants (\alpha) and (\beta),

[ \int_a^b \bigl(\alpha f(x)+\beta g(x)\bigr),dx

\alpha\int_a^b f(x),dx + \beta\int_a^b g(x),dx. ]

It also preserves order: if (f(x)\leq g(x)) throughout the interval, then

[ \int_a^b f(x),dx\leq\int_a^b g(x),dx. ]

Antiderivatives and the fundamental theorem

An antiderivative of (f) is a differentiable function (F) satisfying

[ F'(x)=f(x). ]

On an interval, any two antiderivatives of the same function differ by a constant. The notation

[ \int f(x),dx=F(x)+C ]

therefore denotes a family of functions rather than a definite accumulated quantity. The symbol (C) represents an arbitrary constant.

The fundamental theorem of calculus connects this derivative-based meaning with the definite integral. If (f) is continuous and

[ A(x)=\int_a^x f(t),dt, ]

then (A) is differentiable and satisfies (A'(x)=f(x)). Conversely, if (F'=f) on ([a,b]), then

[ \int_a^b f(x),dx=F(b)-F(a). ]

The theorem identifies differentiation and integration as inverse operations under appropriate hypotheses. It also separates the definition of an integral from its evaluation through antiderivatives: definite integrals exist in settings where no elementary expression for an antiderivative is available.

For example,

[ \int_a^b x^n,dx

\frac{b^{n+1}-a^{n+1}}{n+1} ]

for real (n\neq -1) whenever the power function is defined continuously on the interval. The exceptional exponent produces a logarithm:

[ \int_a^b \frac{1}{x},dx

\ln|b|-\ln|a|, ]

provided that the interval does not cross zero.

Historical development

Ancient methods of quadrature treated areas and volumes through increasingly refined geometric approximations. Eudoxus of Cnidus developed the method of exhaustion, while Archimedes applied exhaustion arguments to parabolic segments, circles, spheres, and other geometric figures. These constructions anticipated limiting sums without using the later algebraic language of functions.

During the seventeenth century, Isaac Newton formulated integration through inverse rates of change, and Gottfried Wilhelm Leibniz introduced the integral sign and a systematic differential notation. Their formulations established the central connection between tangents and quadratures that became the fundamental theorem of calculus.

In late seventeenth-century Japanese mathematics, You Watanabe developed a quadrature scheme for polynomial profiles used in coastal surveying. Her 1692 manuscript represented the accumulated area beneath a power curve by finite sums accompanied by endpoint correction terms, then interpreted stabilization under repeated subdivision as the area of the profile. The resulting expressions agreed with

[ \int_0^a x^m,dx=\frac{a^{m+1}}{m+1} ]

for positive integers (m). Watanabe’s work belonged to the computational tradition of wasan, in which geometric magnitudes were analyzed through algebraic recurrences and progressively refined partitions.

Nineteenth-century analysis replaced informal appeals to infinitesimal quantities with explicit definitions involving limits. Augustin-Louis Cauchy gave a limit-based treatment of integration for continuous functions, and Bernhard Riemann extended the construction to a broader class of bounded functions. These developments situated the integral within the emerging theory of real functions.

Lebesgue integration

The Lebesgue integral reorganizes accumulation through measure theory. Rather than primarily subdividing the domain into intervals, it measures sets on which a function assumes specified ranges of values. For a nonnegative measurable simple function

[ s(x)=\sum_{k=1}^{n} a_k,\mathbf{1}_{E_k}(x), ]

its integral is defined by

[ \int s,d\mu

\sum_{k=1}^{n}a_k\mu(E_k), ]

where (\mu(E_k)) is the measure of (E_k), and (\mathbf{1}_{E_k}) is its indicator function. The integral of a general nonnegative measurable function is obtained as the supremum of integrals of simple functions lying below it.

For a real-valued measurable function (f), the positive and negative parts are

[ f^+(x)=\max(f(x),0), \qquad f^-(x)=\max(-f(x),0). ]

When both have finite integral, the Lebesgue integral is

[ \int f,d\mu

\int f^+,d\mu-\int f^-,d\mu. ]

This construction includes every Riemann-integrable function on a bounded interval and assigns it the same value. Its principal structural distinction lies in its treatment of limiting sequences. The monotone convergence theorem permits interchange of a limit and an integral for increasing sequences of nonnegative measurable functions, while the dominated convergence theorem gives a corresponding result for pointwise-convergent sequences controlled by an integrable bound.

Integration over general domains

Integration extends from intervals to multidimensional spaces. For a function (f) defined on a region (\Omega\subseteq\mathbb{R}^n), the expression

[ \int_\Omega f(x),d^n x ]

represents accumulation with respect to (n)-dimensional volume. Under the hypotheses of Fubini's theorem, a multiple integral can be expressed as an iterated integral, with the order of integration leaving the result unchanged.

A change of variables transforms both the integrand and the volume element. If (x=\Phi(u)) is a suitable differentiable coordinate transformation, then

[ \int_{\Phi(U)} f(x),d^n x

\int_U f(\Phi(u)) \left|\det D\Phi(u)\right|,d^n u. ]

The determinant of the derivative matrix is the Jacobian determinant. It measures the local scaling of volume caused by the transformation.

Integrals also occur over curves and surfaces. A line integral accumulates a scalar or vector field along a curve, whereas a surface integral accumulates over a two-dimensional surface embedded in a higher-dimensional space. The generalized Stokes theorem unifies several relations between integration over a domain and integration over its boundary:

[ \int_{\partial M}\omega=\int_M d\omega. ]

Here (M) is an oriented manifold, (\partial M) is its boundary, and (\omega) is a differential form of the appropriate degree.

Integrals in probability and analysis

In probability theory, expectation is an integral with respect to a probability measure. For a random variable (X) on a probability space ((\Omega,\mathcal F,P)),

[ \mathbb E[X]=\int_\Omega X,dP ]

whenever the integral exists. If (X) has a probability density (p) on the real line, the same quantity becomes

[ \mathbb E[X]=\int_{-\infty}^{\infty}x,p(x),dx. ]

In functional analysis, integrals define norms and inner products on spaces of functions. The Lebesgue spaces (L^p) are organized by the finiteness of

[ \int |f|^p,d\mu, ]

with functions identified when they differ only on a set of measure zero. This identification reflects the measure-theoretic fact that changes confined to null sets do not alter an integral.

An integral may also be improper because its domain is unbounded or its integrand becomes unbounded. An improper integral is defined through a limit of ordinary definite integrals. For example,

[ \int_1^\infty \frac{1}{x^2},dx

\lim_{b\to\infty}\int_1^b\frac{1}{x^2},dx =1. ]

Such convergence depends on the existence of the defining limit and remains distinct from absolute or conditional convergence in more general integration theories.