Real Analysis/Fundamental Theorem of Calculus
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Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is said to be the central theorem of elementary calculus. It states effectively that "Differentiation" and "Integration" are inverse operations.
Conventionally, the theorem is presented in two parts
First Form of the Fundamental Theorem
Let be differentiable on and let for all
Let be Riemann integrable on
Let and let be given.
Then, there exists such that for a partition implies that
Consider a partition and let . By Lagrange's Mean Value Theorem, we have that there exists that satisfies
Let the tagged partition be the partition along with the tags
But we know that and hence, .
As is arbitrary, that is,
Second Form of the Fundamental Theorem
We first define what is known as the "Indefinite Integral"
Let be Riemann integrable on .
We define the Indefinite Integral of to be the function given by
Let be continuous at
Let be the indefinite integral of
Then, is differentiable at and
Let be given, and let but
Observe that (say).
There exists such that if a partition then, (note that in this proof, all the Riemann sums are over the interval ).
As is integrable over , it is bounded over that interval. Hence, let . Thus,
As is continuous at , there exists such that whenever .
That is, , or