definite integral

mathematics

Also known as: Riemann integral

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June 25, 2026

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In mathematics, an integral can refer to two related concepts: a numerical value representing the area under a function's graph over an interval (a definite integral), or a new function whose derivative is the original function (an indefinite integral or antiderivative).12

The definite integral calculates the area bounded by a function's curve, the x-axis, and specific vertical lines. The indefinite integral is not unique because the derivative of a constant is zero.2

The fundamental theorem of calculus is a key principle that connects differentiation and integration. It states that a continuous function has an antiderivative, and the definite integral can be found by calculating the difference between the antiderivative evaluated at the interval's endpoints. This theorem is fundamental to calculus and the physical sciences. The process of finding an indefinite integral is called integration.3415

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analysis
The Riemann integral
) The task of analysis is to provide not a computational method but a sound logical foundation for limiting processes. Oddly enough, when it comes to formalizing the integral, the most difficult part is to define the term area. It is easy to define…
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) The task of analysis is to provide not a computational method but a sound logical foundation for limiting processes. Oddly enough, when it comes to formalizing the integral, the most difficult part is to define the term area. It is easy to define…
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integral
…by the fact that a definite integral of any function that can be integrated can be found using the indefinite integral and a corollary to the fundamental theorem of calculus. The definite integral (also called Riemann integral) of a function f(x) is denoted as(see integration [for symbol]) and is equal
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…by the fact that a definite integral of any function that can be integrated can be found using the indefinite integral and a corollary to the fundamental theorem of calculus. The definite integral (also called Riemann integral) of a function f(x) is denoted as(see integration [for symbol]) and is equal
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analysis
Measure theory
…a number of deficiencies in Riemann’s way of defining the integral. (The Riemann integral is explained in the section Integration.) Many functions with reasonable properties turned out not to possess integrals in Riemann’s sense. Moreover, certain limiting procedures, when applied to sequences not of numbers but of functions, behaved in…
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…a number of deficiencies in Riemann’s way of defining the integral. (The Riemann integral is explained in the section Integration.) Many functions with reasonable properties turned out not to possess integrals in Riemann’s sense. Moreover, certain limiting procedures, when applied to sequences not of numbers but of functions, behaved in…
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Henri-Léon Lebesgue
…mathematician whose generalization of the Riemann integral revolutionized the field of integration.
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…mathematician whose generalization of the Riemann integral revolutionized the field of integration.
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