Integral (Mathematics)
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Its inverse operation is differentiation, and the connection between the two is given by the fundamental theorem of calculus.
Integration is one of the two main operations in calculus. While differentiation studies instantaneous rates of change, integration accumulates quantities over continuous intervals.
Definition
The most common definition of the integral used in introductory calculus is the Riemann integral, introduced by Carl Friedrich Gauss and rigorously defined by Bernhard Riemann in 1854. It is defined as the limit of sums of areas of rectangles that approximate the area under a curve.
Formally, if $f$ is a real-valued function defined on a closed interval $[a, b]$, the definite integral is written as:
where $\Delta x = \frac{b-a}{n}$ and $x_i^*$ is any point in the $i$-th subinterval. If this limit exists, $f$ is said to be Riemann integrable on $[a, b]$.
Notation
The standard notation for the integral was introduced by Gottfried Wilhelm Leibniz in 1675. The integral sign $\int$ is an elongated 'S', standing for summa (sum). The basic structure of the notation is:
Here, $f(x)$ is the integrand, $dx$ indicates the variable of integration, and in the definite case, the numbers at the top and bottom of the integral sign are the upper and lower limits of integration, respectively.
Types of Integrals
Indefinite Integral
An indefinite integral represents a family of functions whose derivatives are equal to the integrand. It is also called the antiderivative:
where $F'(x) = f(x)$ and $C$ is an arbitrary constant called the constant of integration.
Definite Integral
The definite integral computes the net signed area between the function and the $x$-axis over an interval $[a, b]$:
Part 1: If $f$ is continuous on $[a, b]$ and $F(x) = \int_{a}^{x} f(t)\,dt$, then $F'(x) = f(x)$.
Part 2: If $F$ is an antiderivative of $f$, then $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$.
Improper Integrals
These extend the concept to unbounded intervals or functions with vertical asymptotes:
Lebesgue Integral
Introduced by Henri Lebesgue in 1901, this measure-theoretic approach generalizes the Riemann integral, allowing integration of a broader class of functions and providing powerful convergence theorems essential in modern analysis, probability, and quantum mechanics.
Properties
Integrals satisfy several fundamental algebraic and analytical properties:
- Linearity: $\int [c f(x) + d g(x)] \, dx = c\int f(x)\,dx + d\int g(x)\,dx$
- Additivity: $\int_{a}^{b} f(x)\,dx + \int_{b}^{c} f(x)\,dx = \int_{a}^{c} f(x)\,dx$
- Reversal: $\int_{a}^{b} f(x)\,dx = -\int_{b}^{a} f(x)\,dx$
- Monotonicity: If $f(x) \geq g(x)$ on $[a,b]$, then $\int_{a}^{b} f(x)\,dx \geq \int_{a}^{b} g(x)\,dx$
- MVT for Integrals: If $f$ is continuous on $[a,b]$, there exists $c \in (a,b)$ such that $\int_{a}^{b} f(x)\,dx = f(c)(b-a)$
Applications
Integration is foundational across mathematics, physics, engineering, economics, and data science:
- Geometry: Computing areas, volumes of revolution, arc lengths, and surface areas
- Physics: Work, center of mass, moment of inertia, fluid pressure, and electromagnetic fields
- Probability: Probability density functions, expected values, and cumulative distributions
- Economics: Consumer surplus, producer surplus, and aggregate quantities from marginal functions
- Engineering: Signal processing, control systems, and thermodynamic cycle analysis
See Also
References
- Apostol, T. M. (1974). Mathematical Analysis (2nd ed.). Addison-Wesley.
- Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill.
- Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications. Wiley.
- Weisstein, E. W. "Integral." From MathWorld. Wolfram Research.