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Integral Calculus: A Brief Overview

Integration is the process of computing an integral, which is one of the two fundamental operations of calculus, the other being differentiation.
Integrals are used to calculate areas, volumes, and their generalizations in mathematics and physics.
The integrals enumerated here are definite integrals, which can be interpreted as 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.
Bernhard Riemann 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.
Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed.
The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus (ca. 370 BC).
The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Leibniz and Newton.
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.
Integration was first rigorously formalized, using limits, by Riemann.
The notation for the indefinite integral was introduced by Gottfried Wilhelm Leibniz in 1675.
There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions.Overview of Integrals
Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval.
The Lebesgue integral allows a wider class of functions to be integrated and begins with a measure, μ.
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.
Although the Riemann and Lebesgue integrals are the most widely used definitions of the integral, a number of others exist.
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.
The set of real-valued Lebesgue-integrable functions on a given measure space E with measure μ is closed under taking linear combinations and form a vector space.
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).
The fundamental theorem of calculus is the statement that differentiation and integration are inverse operations.
Improper integrals occur when one or more of the conditions of a proper Riemann integral is not satisfied.
The double integral of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane that contains its domain.
The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces inside higher-dimensional spaces.
A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve.
A surface integral generalizes double integrals to integration over a surface.Overview of Integrals
Integrals are mathematical tools used to find the area under a curve, the volume of a solid, or the work done by a force, among other things.
There are two types of integrals: indefinite integrals and definite integrals.
Indefinite integrals are those that do not have limits of integration and are used to find the antiderivative of a function.
Definite integrals, on the other hand, have limits of integration and are used to find the area or volume between a curve and an axis or between two curves.
The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus.
Techniques include integration by substitution, integration by parts, integration by trigonometric substitution, and integration by partial fractions.
Extensive tables of integrals have been compiled and published over the years, but many nonelementary integrals cannot be expressed in closed form involving only elementary functions.
Symbolic integration has been one of the motivations for the development of computer algebra systems that are specifically designed to perform difficult or tedious tasks, including integration.
Definite integrals may be approximated using several methods of numerical integration, such as the Trapezoidal Rule, Simpson's Rule, and Gaussian Quadrature.
Applications of integrals include probability theory, physics, thermodynamics, and engineering.
Integrals of differential forms and surface integrals have applications in physics, particularly with the classical theory of electromagnetism.
The exterior derivative plays the role of the gradient and curl of vector calculus, and Stokes' theorem simultaneously generalizes the three theorems of vector calculus: the divergence theorem, Green's theorem, and the Kelvin-Stokes theorem.
The discrete equivalent of integration is summation. Summations and integrals can be put on the same foundations using the theory of Lebesgue integrals or time-scale calculus.Methods of Integration
The rectangle method and trapezoidal rule are used to approximate the integral of a function.
Simpson’s rule is a more accurate approximation method that uses piecewise quadratic functions.
The Newton-Cotes formulas are a family of quadrature rules that approximate the polynomial on each subinterval by a degree n polynomial.
Clenshaw-Curtis quadrature is used to avoid numerical inaccuracy due to Runge’s phenomenon.
Romberg’s method halves the step widths incrementally to give trapezoid approximations.
Gaussian quadrature evaluates the function at the roots of a set of orthogonal polynomials.
Monte Carlo integration is used for the computation of higher-dimensional integrals.
The area of an arbitrary two-dimensional shape can be determined using a measuring instrument called planimeter.
The volume of irregular objects can be measured with precision by the fluid displaced as the object is submerged.
The area can sometimes be found via geometrical compass-and-straightedge constructions of an equivalent square.
Integration by differentiation can be used to calculate an integral by means of differentiation.
The fundamental theorem of calculus allows for straightforward calculations of basic functions.