**Integral**

**Integration** is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus. Given a function *f* of a real variable *x* and an interval of the real line, the **definite integral**

is defined informally to be the area of the region in the *xy*-plane bounded by the graph of *f*, the *x*-axis, and the vertical lines *x* = *a* and *x* = *b*, such that area above the *x*-axis adds to the total, and that below the *x*-axis subtracts from the total.

The term *integral* may also refer to the notion of the antiderivative, a function *F* whose derivative is the given function *f*. In this case, it is called an *indefinite integral* and is written:

The integrals discussed in this article are termed *definite integrals*.

The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if *f* is a continuous real-valued function defined on a closed interval, then, once an antiderivative *F* of *f* is known, the definite integral of *f* over that interval is given by

Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. The founders of the calculus thought of the integral as an infinite sum of rectangles of infinitesimal width. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. Integrals of differential forms play a fundamental role in modern differential geometry. These generalizations of integrals first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics. There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.

Read more about Integral: Terminology and Notation, Introduction, Formal Definitions, Fundamental Theorem of Calculus, Some Important Definite Integrals

### Famous quotes containing the word integral:

“Make the most of your regrets; never smother your sorrow, but tend and cherish it till it come to have a separate and *integral* interest. To regret deeply is to live afresh.”

—Henry David Thoreau (1817–1862)

“... no one who has not been an *integral* part of a slaveholding community, can have any idea of its abominations.... even were slavery no curse to its victims, the exercise of arbitrary power works such fearful ruin upon the hearts of slaveholders, that I should feel impelled to labor and pray for its overthrow with my last energies and latest breath.”

—Angelina Grimké (1805–1879)

“Painting myself for others, I have painted my inward self with colors clearer than my original ones. I have no more made my book than my book has made me—a book consubstantial with its author, concerned with my own self, an *integral* part of my life; not concerned with some third-hand, extraneous purpose, like all other books.”

—Michel de Montaigne (1533–1592)