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'fundamental theorems of calculus for Lebesgue integration'
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| Title of object: |
fundamental theorems of calculus for Lebesgue integration |
| Canonical Name: |
FundamentalTheoremOfCalculus |
| Type: |
Theorem |
| Created on: |
2002-02-24 19:50:49 |
| Modified on: |
2004-03-03 14:46:47 |
| Classification: |
msc:26-00 |
| Synonyms: |
fundamental theorems of calculus for Lebesgue integration=first fundamental theorem of calculus
fundamental theorems of calculus for Lebesgue integration=second fundamental theorem of calculus
fundamental theorems of calculus for Lebesgue integration=fundamental theorem of calculus |
Revision comment (for changes between this and next version):
| Changes for correction #3727 ('Second Form ==> First Form'). |
Preamble:
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Content:
Loosely, the \emph{Fundamental Theorems of Calculus} serve to demonstrate that integration and differentiation are inverse processes. Suppose that $F(x)$ is an absolutely continuous function on an interval $[a,b]\subset\mathbb{R}$.
{\bf First Fundamental Theorem:}
There exists a function $f(t)$ Lebesgue-integrable on $[a,b]$ such that for any $x\in [a,b]$, we have $F(x)-F(a)=\int_a^x f(t) dt$.
{\bf Second Fundamental Theorem:}
$F(x)$ is differentiable almost everywhere on $[a,b]$ and its derivative, denoted $F'(x)$, is Lebesgue-integrable on that inteval. In addition, we have the relation $F(x)-F(a)=\int_a^x F'(t)dt$ for any $x\in [a,b]$. |
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