fundamental theorem of calculus
Let be a continuous function, let be givenand consider the integral function defined on as
Then is an antiderivative of that is, is differentiable in and
The previous relation rewritten as
shows that the differentiation operator is the inverse
of the integration operator . This formula
is sometimes called Newton-Leibniz formula.
On the other hand if is a continuous functionand is any antiderivative of , i.e. for all , then
(1) |
This shows that up to a constant, the integration operator is the inverse of the derivative operator:
Notes
Equation (1) is sometimes called “Barrow’s rule” or “Barrow’s formula”.
Title | fundamental theorem of calculus![]() |
Canonical name | FundamentalTheoremOfCalculus |
Date of creation | 2013-03-22 14:13:27 |
Last modified on | 2013-03-22 14:13:27 |
Owner | paolini (1187) |
Last modified by | paolini (1187) |
Numerical id | 13 |
Author | paolini (1187) |
Entry type | Theorem |
Classification | msc 26A42 |
Synonym | Newton-Leibniz |
Synonym | Barrow’s rule |
Synonym | Barrow’s formula |
Related topic | FundamentalTheoremOfCalculus |
Related topic | FundamentalTheoremOfCalculusForKurzweilHenstockIntegral |
Related topic | FundamentalTheoremOfCalculusForRiemannIntegration |
Related topic | LaplaceTransformOfFracftt |
Related topic | LimitsOfNaturalLogarithm |
Related topic | FundamentalTheoremOfIntegralCalculus |