Integral

In mathematics, an integral is the continuous analog of a sum, and is used to calculate areas, volumes, and their generalizations. The process of computing an integral, called integration, is one of the two fundamental operations of calculus, along with differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter.

A definite integral computes the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integration to differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations.

Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral; it is more general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.

Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting two points in space. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.

The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek mathematicians Democritus and Eudoxus (4th century BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate the area of a circle, the surface area and volume of a sphere, area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.

A similar method was independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere.

In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen (c. 965 – c. 1040 AD), extended Archimedes' method of exhaustion, finding the volume of the solid of revolution formed by rotating a parabola around a line perpendicular to its axis. To achieve this, Al-Haytham invented a way of computing the sums of kth powers, which he applied for the sums of squares and the sums of fourth powers.

The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of Cavalieri with his method of indivisibles, and work by Fermat, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of x up to degree n = 9 in Cavalieri's quadrature formula. The case n = −1 required the invention of a function, the hyperbolic logarithm, achieved by quadrature of the hyperbola in 1647.

Further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers.

The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions with continuous domains. This framework eventually became modern calculus, whose notation for integrals is drawn directly from the work of Leibniz.

While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them "ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits. Integration was first rigorously formalized, using limits, by Riemann. Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered, particularly in the context of Fourier analysis, to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield of real analysis). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system.

The notation for the indefinite integral was introduced by Gottfried Wilhelm Leibniz in 1675. He adapted the integral symbol, ∫, from the letter ſ (long s), standing for summa (written as ſumma; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819-1820, reprinted in his book of 1822.

Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with .x or x′, which are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.

In general, the integral of a real-valued function f(x) with respect to a real variable x on an interval [a, b] is written as

\(\int_{a}^{b} f(x) \,dx.\)

The integral symbol ∫ represents integration. The symbol dx, called the differential of the variable x, indicates that the variable of integration is x. The function f(x) is called the integrand, the points a and b are called the limits (or bounds) of integration, and the integral is said to be over the interval [a, b], called the interval of integration. A function is said to be integrable if its integral is well-defined and finite. (The type of integral being considered determines the type of integrability this means, thus Riemann integrable means that the upper and lower Riemann sums converge over the domain, Lebesgue integrable means that the Lebesgue integral exists and is finite, and so on.) If limits are specified, the integral is called a definite integral.

When the limits are omitted, as in

\(\int f(x) \,dx,\)

the integral is called an indefinite integral, which represents a class of functions (the antiderivative) whose derivative is the integrand. The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article).

In advanced settings, it is not uncommon to leave out dx when only the simple Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write \(\int_a^b (c_1f+c_2g) = c_1\int_a^b f + c_2\int_a^b g\) to express the linearity of the integral, a property shared by the Riemann integral and all generalizations thereof.

Integrals appear in many practical situations. For instance, from the length, width and depth of a swimming pool which is rectangular with a flat bottom, one can determine the volume of water it can contain, the area of its surface, and the length of its edge. But if it is oval with a rounded bottom, integrals are required to find exact and rigorous values for these quantities. In each case, one may divide the sought quantity into infinitely many infinitesimal pieces, then sum the pieces to achieve an accurate approximation.

As another example, to find the area of the region bounded by the graph of the function f(x) = \(\sqrt{x}\) between x = 0 and x = 1, one can divide the interval into five pieces (0, 1/5, 2/5, ..., 1), then construct rectangles using the right end height of each piece (thus √0, √1/5, √2/5, ..., √1) and sum their areas to get the approximation

\(\textstyle \sqrt{\frac{1}{5}}\left(\frac{1}{5}-0\right)+\sqrt{\frac{2}{5}}\left(\frac{2}{5}-\frac{1}{5}\right)+\cdots+\sqrt{\frac{5}{5}}\left(\frac{5}{5}-\frac{4}{5}\right)\approx 0.7497,\)

which is larger than the exact value. Alternatively, when replacing these subintervals by ones with the left end height of each piece, the approximation one gets is too low: with twelve such subintervals the approximated area is only 0.6203. However, when the number of pieces increases to infinity, it will reach a limit which is the exact value of the area sought (in this case, 2/3). One writes

\(\int_{0}^{1} \sqrt{x} \,dx = \frac{2}{3},\)

which means 2/3 is the result of a weighted sum of function values, √x, multiplied by the infinitesimal step widths, denoted by dx, on the interval [0, 1].

There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but are also occasionally for pedagogical reasons. The most commonly used definitions are Riemann integrals and Lebesgue integrals.

The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. A tagged partition of a closed interval [a, b] on the real line is a finite sequence

\(a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!\)

This partitions the interval [a, b] into n sub-intervals [x_i−1, x_i] indexed by i, each of which is "tagged" with a specific point t_i ∈ [x_i−1, x_i]. A Riemann sum of a function f with respect to such a tagged partition is defined as

\(\sum_{i=1}^n f(t_i) \, \Delta_i ;\)

thus each term of the sum is the area of a rectangle with height equal to the function value at the chosen point of the given sub-interval, and width the same as the width of sub-interval, Δ_i = x_i−x_i−1. The mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, max_i=1...n Δ_i. The Riemann integral of a function f over the interval [a, b] is equal to S if:

For all \(\varepsilon > 0\) there exists \(\delta > 0\) such that, for any tagged partition \([a, b]\) with mesh less than \(\delta\),

\(\left| S - \sum_{i=1}^n f(t_i) \, \Delta_i \right| < \varepsilon.\)

When the chosen tags are the maximum (respectively, minimum) value of the function in each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral.

It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral. Therefore, it is of great importance to have a definition of the integral that allows a wider class of functions to be integrated.

Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus Henri Lebesgue introduced the integral bearing his name, explaining this integral thus in a letter to Paul Montel:

As Folland puts it, "To compute the Riemann integral of f, one partitions the domain [a, b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f ". The definition of the Lebesgue integral thus begins with a measure, μ. In the simplest case, the Lebesgue measure μ(A) of an interval A = [a, b] is its width, b − a, so that the Lebesgue integral agrees with the (proper) Riemann integral when both exist. In more complicated cases, the sets being measured can be highly fragmented, with no continuity and no resemblance to intervals.

Using the "partitioning the range of f " philosophy, the integral of a non-negative function f : R → R should be the sum over t of the areas between a thin horizontal strip between y = t and y = t + dt. This area is just μ{ x : f(x) > t} dt. Let f(t) = μ{ x : f(x) > t }. The Lebesgue integral of f is then defined by

\(\int f = \int_0^\infty f^*(t)\,dt\)

where the integral on the right is an ordinary improper Riemann integral (f is a strictly decreasing positive function, and therefore has a well-defined improper Riemann integral). For a suitable class of functions (the measurable functions) this defines the Lebesgue integral.

A general measurable function f is Lebesgue-integrable if the sum of the absolute values of the areas of the regions between the graph of f and the x-axis is finite:

\(\int_E |f|\,d\mu < + \infty.\)

In that case, the integral is, as in the Riemannian case, the difference between the area above the x-axis and the area below the x-axis:

\(\int_E f \,d\mu = \int_E f^+ \,d\mu - \int_E f^- \,d\mu\)

Condensed: the full section is in Wikipedia.

Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist, including:

• The Darboux integral, which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to the Riemann integral. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of being easier to define than Riemann integrals.
• The Riemann-Stieltjes integral, an extension of the Riemann integral which integrates with respect to a function as opposed to a variable.
• The Lebesgue-Stieltjes integral, further developed by Johann Radon, which generalizes both the Riemann-Stieltjes and Lebesgue integrals.
• The Daniell integral, which subsumes the Lebesgue integral and Lebesgue-Stieltjes integral without depending on measures.
• The Haar integral, used for integration on locally compact topological groups, introduced by Alfréd Haar in 1933.
• The Henstock-Kurzweil integral, variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock.
• The Khinchin integral, named after Aleksandr Khinchin.
• The Itô integral and Stratonovich integral, which define integration with respect to semimartingales such as Brownian motion.
• The Young integral, which is a kind of Riemann-Stieltjes integral with respect to certain functions of unbounded variation.
• The rough path integral, which is defined for functions equipped with some additional "rough path" structure and generalizes stochastic integration against both semimartingales and processes such as the fractional Brownian motion.
• The Choquet integral, a subadditive or superadditive integral created by Gustave Choquet in 1953.
• The Bochner integral, a generalization of the Lebesgue integral to functions that take values in a Banach space.

The collection of Riemann-integrable functions on a closed interval [a, b] forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration

\(f \mapsto \int_a^b f(x) \; dx\)

is a linear functional on this vector space. Thus, the collection of integrable functions is closed under taking linear combinations, and the integral of a linear combination is the linear combination of the integrals:

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

Similarly, the set of real-valued Lebesgue-integrable functions on a given measure space E with measure μ is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral

\(f\mapsto \int_E f \, d\mu\)

is a linear functional on this vector space, so that:

\(\int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu.\)

More generally, consider the vector space of all measurable functions on a measure space (E,μ), taking values in a locally compact complete topological vector space V over a locally compact topological field K, f : E → V. Then one may define an abstract integration map assigning to each function f an element of V or the symbol ∞,

\(f\mapsto\int_E f \,d\mu, \,\)

that is compatible with linear combinations. In this situation, the linearity holds for the subspace of functions whose integral is an element of V (i.e. "finite"). The most important special cases arise when K is R, C, or a finite extension of the field Q_p of p-adic numbers, and V is a finite-dimensional vector space over K, and when K = C and V is a complex Hilbert space.

Linearity, together with some natural continuity properties and normalization for a certain class of "simple" functions, may be used to give an alternative definition of the integral. This is the approach of Daniell for the case of real-valued functions on a set X, generalized by Nicolas Bourbaki to functions with values in a locally compact topological vector space. See Hildebrandt 1953 for an axiomatic characterization of the integral.

A number of general inequalities hold for Riemann-integrable functions defined on a closed and bounded interval [a, b] and can be generalized to other notions of integral (Lebesgue and Daniell).

• Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that m ≤ f (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(b − a) and M(b − a), it follows that \[m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a).\]
• Inequalities between functions. If f(x) ≤ g(x) for each x in [a, b] then each of the upper and lower sums of f is bounded above by the upper and lower sums, respectively, of g. Thus \[\int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx.\] This is a generalization of the above inequalities, as M(b − a) is the integral of the constant function with value M over [a, b]. In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if f(x) < g(x) for each x in [a, b], then \[\int_a^b f(x) \, dx < \int_a^b g(x) \, dx.\]
• Subintervals. If [c, d] is a subinterval of [a, b] and f (x) is non-negative for all x, then \[\int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx.\]
• Products and absolute values of functions. If f and g are two functions, then we may consider their pointwise products and powers, and absolute values: \[(fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; |f| (x) = |f(x)|.\] If f is Riemann-integrable on [a, b] then the same is true for |f|, and \[\left| \int_a^b f(x) \, dx \right| \leq \int_a^b | f(x) | \, dx.\] Moreover, if f and g are both Riemann-integrable then fg is also Riemann-integrable, and \[\left( \int_a^b (fg)(x) \, dx \right)^2 \leq \left( \int_a^b f(x)^2 \, dx \right) \left( \int_a^b g(x)^2 \, dx \right).\] This inequality, known as the Cauchy-Schwarz inequality, plays a prominent role in Hilbert space theory, where the left hand side is interpreted as the inner product of two square-integrable functions f and g on the interval [a, b].
• Hölder's inequality. Suppose that p and q are two real numbers, 1 ≤ p, q ≤ ∞ with ⁠1/p⁠ + ⁠1/q⁠ = 1, and f and g are two Riemann-integrable functions. Then the functions |f| and |g| are also integrable and the following Hölder's inequality holds: \[\left|\int f(x)g(x)\,dx\right| \leq \left(\int \left|f(x)\right|^p\,dx \right)^{1/p} \left(\int\left|g(x)\right|^q\,dx\right)^{1/q}.\] For p = q = 2, Hölder's inequality becomes the Cauchy-Schwarz inequality.
• Minkowski inequality. Suppose that p ≥ 1 is a real number and f and g are Riemann-integrable functions. Then | f |, | g | and | f + g | are also Riemann-integrable and the following Minkowski inequality holds: \[\left(\int \left|f(x)+g(x)\right|^p\,dx \right)^{1/p} \leq \left(\int \left|f(x)\right|^p\,dx \right)^{1/p} + \left(\int \left|g(x)\right|^p\,dx \right)^{1/p}.\] An analogue of this inequality for Lebesgue integral is used in construction of L spaces.

In this section, f is a real-valued Riemann-integrable function. The integral

\(\int_a^b f(x) \, dx\)

over an interval [a, b] is defined if a < b. This means that the upper and lower sums of the function f are evaluated on a partition a = x₀ ≤ x₁ ≤ . . . ≤ x_n = b whose values x_i are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating f within intervals [x_i , x_i +1] where an interval with a higher index lies to the right of one with a lower index. The values a and b, the end-points of the interval, are called the limits of integration of f. Integrals can also be defined if a > b:

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

With a = b, this implies:

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

The first convention is necessary in consideration of taking integrals over subintervals of [a, b]; the second says that an integral taken over a degenerate interval, or a point, should be zero. One reason for the first convention is that the integrability of f on an interval [a, b] implies that f is integrable on any subinterval [c, d], but in particular integrals have the property that if c is any element of [a, b], then:

\(\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.\)

With the first convention, the resulting relation

\(\begin{align} \int_a^c f(x) \, dx &{}= \int_a^b f(x) \, dx - \int_c^b f(x) \, dx \\ &{} = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx \end{align}\)

is then well-defined for any cyclic permutation of a, b, and c.