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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.