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 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 function f , an antiderivative or indefinite integral F may be obtained as the integral of f over an interval with a variable upper bound. This implies the existence of antiderivatives for continuous functions.[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.

Contents
1 History
2 Geometric meaning
3 Physical intuition
4 Formal statements
4.1 First part
4.2 Corollary
4.3 Second part
5 Proof of the first part
6 Proof of the corollary
7 Proof of the second part
8 Relationship between the parts
9 Examples
9.1 Computing a particular integral
9.2 Using the first part
9.3 An integral where the corollary is insufficient
9.4 Theoretical example
10 Generalizations
11 See also