fundamental theorem of calculus
Let f:[a,b]βπ be a continuous function, let cβ[a,b] be given
and consider the integral function F defined on [a,b] as
F(x)=β«xcf(t)πt. |
Then F is an antiderivative of f that is,
F is differentiable in [a,b] and
Fβ²(x)=f(x)ββ |
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 function and 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 βBarrowβs ruleβ or βBarrowβs formulaβ.
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 | Barrowβs rule |
Synonym | Barrowβs formula |
Related topic | FundamentalTheoremOfCalculus |
Related topic | FundamentalTheoremOfCalculusForKurzweilHenstockIntegral |
Related topic | FundamentalTheoremOfCalculusForRiemannIntegration |
Related topic | LaplaceTransformOfFracftt |
Related topic | LimitsOfNaturalLogarithm |
Related topic | FundamentalTheoremOfIntegralCalculus |