Integration is, alongside [differentiation](/page/Derivative), one of the two fundamental operations of calculus. Where differentiation extracts the instantaneous rate of change of a function, integration reconstructs a quantity from its rate of change — it computes areas, volumes, total mass, accumulated work, and probabilities. The [Fundamental Theorem of Calculus](/theorems/632) reveals that these two operations are inverses: differentiation undoes integration, and integration undoes differentiation.

But the concept of "integral" is not a single definition — it is a family of increasingly powerful constructions, each designed to handle a larger class of [functions](/page/Function). The [Riemann integral](/page/Riemann%20Integral) (partitioning the domain into subintervals) suffices for [continuous](/page/Continuous) functions and many applications in calculus and physics. The [Lebesgue integral](/page/Lebesgue%20Integral) (partitioning the range and measuring the preimages) handles a far larger class — all measurable functions — and supports the powerful convergence theorems that analysis requires. Beyond these, the Stieltjes integral, the Henstock-Kurzweil integral, the integral with respect to a general measure, and the integral on manifolds each extend the concept further.

This page develops the integral at the conceptual level: the problem it solves, the key ideas behind the Riemann and Lebesgue approaches, the properties that any reasonable integral must satisfy, the Fundamental Theorem of Calculus, the convergence theorems, and integration in higher dimensions and on general spaces. The detailed constructions are developed on dedicated pages.

[motivation] ## Motivation ### The Area Problem The oldest motivation for integration is the computation of area. The area of a rectangle with sides $a$ and $b$ is $ab$ — but what is the area of the region bounded by a curve $y = f(x)$, the $x$-axis, and the vertical lines $x = a$ and $x = b$? Archimedes answered this for parabolas using the method of exhaustion: approximate the region by inscribed and circumscribed polygons, show the two approximations converge to the same value, and declare that value to be the area. This is, in essence, the Riemann integral — the area is the [limit](/page/Limit) of sums of rectangular areas as the rectangles become infinitely thin. ### Beyond Area: Accumulation Integration is far more than an area computation. Whenever a quantity accumulates at a varying rate, the total accumulation is an integral. If $v(t)$ is velocity, then $\int_a^b v(t) \, d\mathcal{L}^1(t)$ is displacement. If $\rho(x)$ is mass density, then $\int_U \rho(x) \, d\mathcal{L}^n(x)$ is total mass. If $f(x)$ is a probability density, then $\int_A f(x) \, d\mathcal{L}^1(x)$ is the probability of the event $A$. In each case, the integral "adds up" infinitesimal contributions to produce a finite total. ### Why Multiple Theories? The Riemann integral — defined by partitioning the $x$-axis and summing $f(x_i^*) \Delta x_i$ — works well for continuous functions but fails for many functions that arise naturally in analysis. The characteristic function $\mathbb{1}_\mathbb{Q}$ of the rationals (equal to $1$ on $\mathbb{Q}$ and $0$ on $\mathbb{R} \setminus \mathbb{Q}$) is not Riemann integrable, because every subinterval contains both rationals and irrationals, so the upper and lower Riemann sums never agree. The pointwise limit of a [sequence](/page/Sequence) of Riemann-integrable functions need not be Riemann integrable. And the Riemann integral does not interact well with [limits](/page/Limit): there exist sequences $f_n \to f$ pointwise with $\int f_n \not\to \int f$. The Lebesgue integral resolves all these difficulties by replacing the domain partition with a range partition: instead of asking "what is $f$ on the interval $[x_i, x_{i+1}]$?", ask "how much of the domain does $f$ map to values near $y$?" This requires measuring the "size" of sets (measure theory) but produces an integral that handles all measurable functions, supports the [Dominated Convergence Theorem](/theorems/4), and provides the foundation for probability theory, functional analysis, and PDE theory. [/motivation]

## Definition

There is no single "definition of the integral" — each theory of integration provides its own construction. However, all reasonable integrals share a common structure: the integral is a **linear functional** on a space of functions that is **monotone** (nonnegative functions have nonnegative integrals) and satisfies an appropriate **convergence theorem**.

### The Abstract Viewpoint

At the most abstract level, an integral on a [set](/page/Set) $X$ with respect to a measure $\mu$ is a linear functional

\begin{align*} I: \mathcal{F} \to \mathbb{R} \quad (\text{or } \mathbb{C}), \qquad f \mapsto I(f) = \int_X f \, d\mu, \end{align*}

defined on some vector space $\mathcal{F}$ of functions $f: X \to \mathbb{R}$, satisfying:

1. **Linearity:** $I(\alpha f + \beta g) = \alpha I(f) + \beta I(g)$. 2. **Monotonicity:** if $f \ge 0$ a.e., then $I(f) \ge 0$. 3. **Normalisation:** $I(\mathbb{1}_{[0,1]}) = 1$ (or more generally, $I(\mathbb{1}_E) = \mu(E)$ for measurable sets $E$).

The different theories of integration differ in the choice of $\mathcal{F}$ (which functions are integrable?), the construction of $I$ (how is the integral computed?), and the convergence theorems (when does $\int f_n \to \int f$?).

[definition:Integral With Respect To A Measure] Let $(X, \Sigma, \mu)$ be a measure space (where $X$ is a set, $\Sigma$ is a $\sigma$-algebra of measurable subsets, and $\mu: \Sigma \to [0, \infty]$ is a measure). For a measurable function $f: X \to [0, \infty]$, the **Lebesgue integral** of $f$ with respect to $\mu$ is \begin{align*} \int_X f \, d\mu := \sup\left\{\int_X s \, d\mu : 0 \le s \le f,\; s \text{ simple}\right\}, \end{align*} where a simple function $s = \sum_{k=1}^n c_k \mathbb{1}_{E_k}$ (with $E_k \in \Sigma$ and $c_k \ge 0$) has integral $\int_X s \, d\mu = \sum_{k=1}^n c_k \mu(E_k)$. For a general measurable function $f: X \to \mathbb{R}$, write $f = f^+ - f^-$ where $f^+ = \max(f, 0)$ and $f^- = \max(-f, 0)$. Then $f$ is **integrable** (written $f \in L^1(X, \mu)$) if both $\int f^+ \, d\mu < \infty$ and $\int f^- \, d\mu < \infty$, and \begin{align*} \int_X f \, d\mu := \int_X f^+ \, d\mu - \int_X f^- \, d\mu. \end{align*} [/definition]

This is the Lebesgue integral, which subsumes the Riemann integral for all functions where both are defined. When $X = [a, b]$, $\Sigma = \mathcal{B}([a, b])$ (the Borel $\sigma$-algebra), and $\mu = \mathcal{L}^1$ (Lebesgue measure), the notation $\int_a^b f(x) \, d\mathcal{L}^1(x)$ or simply $\int_a^b f(x) \, dx$ is used.

## The Riemann and Lebesgue Approaches

The two classical approaches to integration partition different things. The Riemann integral partitions the *domain*; the Lebesgue integral partitions the *range*. This conceptual difference is the source of all the technical differences between the two theories.

### The Riemann Integral

The Riemann integral of $f: [a, b] \to \mathbb{R}$ is defined by partitioning $[a, b]$ into subintervals $[x_{i-1}, x_i]$, forming the upper sums $U(f, P) = \sum (\sup_{[x_{i-1}, x_i]} f) \Delta x_i$ and lower sums $L(f, P) = \sum (\inf_{[x_{i-1}, x_i]} f) \Delta x_i$, and declaring $f$ Riemann integrable if $\inf_P U(f, P) = \sup_P L(f, P)$.

The Riemann integral handles all [continuous](/page/Continuous) functions (by the [Heine-Cantor theorem](/theorems/280), [continuity](/page/Continuity) on $[a, b]$ is uniform, so the upper and lower sums converge), all monotone functions, and more generally all bounded functions with a set of discontinuities of Lebesgue measure zero (the Lebesgue criterion). Its limitation is that it requires the oscillation of $f$ on small intervals to vanish — a condition that fails for highly discontinuous functions like $\mathbb{1}_\mathbb{Q}$.

### The Lebesgue Integral