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Fundamental theorem of calculus

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each point in time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area.

The first part of the theorem, the first fundamental theorem of calculus, states that for a continuous function f , an antiderivative or indefinite integral F can be obtained as the integral of f over an interval with a variable upper bound.[1]


Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral of a function f over a fixed interval is equal to the change of any antiderivative F between the ends of the interval. This greatly simplifies the calculation of a definite integral provided an antiderivative can be found by symbolic integration, thus avoiding numerical integration.

Examples[edit]

Computing a particular integral[edit]

Suppose the following is to be calculated:

Differentiation under the integral sign

Telescoping series

Fundamental theorem of calculus for line integrals

Notation for differentiation

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Fundamental theorem of calculus"

at Convergence

James Gregory's Euclidean Proof of the Fundamental Theorem of Calculus

Isaac Barrow's proof of the Fundamental Theorem of Calculus

Fundamental Theorem of Calculus at imomath.com

Alternative proof of the fundamental theorem of calculus

MIT.

Fundamental Theorem of Calculus

Mathworld.

Fundamental Theorem of Calculus