The Fundamental Theorem of Calculus
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Definition, upper and lower integrals
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Let M and m denote the supremum and infinum of [Equation] respectively in the interval [Equation]. Let D represent a dissection of [Equation]. Let [Equation] and [Equation] be the sums the upper and lower approximative sums for [Equation] over the dissection D. As before, let [Equation] on the rth interval, [Equation]. We have [Equation] hence [Equation] This is true whatever the dissection. So the set of all dissections D of [Equation] is bounded above and below; hence has a supremum J and an infinum j. We need to show that [Equation]. To do this we first require the following lemma. Lemma If D is a dissection of [Equation] then the introduction of an additional point of division into D decreases the upper sum [Equation]. Proof Let [Equation] be the upper sum for the dissection D. Let the dissection be obtained from the dissection D by the introduction of a additional point of division [Equation] into the interval [Equation]. Let [Equation] on the interval [Equation]; let[Equation] on the interval [Equation] let [Equation] on the interval [Equation]; then [Equation]. Then [Equation] Hence [Equation] Similarly, [Equation] In this case the dissection [Equation] is said to be a refinement of the dissection D. Theorem Suppose D and [Equation] are two dissections of [Equation] and [Equation] is a refinement of both of them. Then [Equation] But [Equation] From these inequalities we obtain [Equation] Which is true for all dissections [Equation]. Hence [Equation] In practice, this inequality may be exact or inexact. Let the function f on the interval [Equation] [Equation] Then every dissection of this function has supremum 1 and infinum 0. It is not sufficient that a function be bounded to entail that the sequence of suprema and infima generated by successive dissections converge to the same limit. When they do converge, the function f is said to be integrable.
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Contents of
The Fundamental Theorem of Calculus
1 Text: The fundamental theorem of calculus
2 Fundamental Theorem of Calculus
3 Infinite series
4 Integration as the reverse of differentiation
5 Convergence and divergence of series
6 Integration as the Sum of Approximations
7 The dissection
8 Properties of infinite series
9 Criteria for convergence
10 Definition, dissection
11 Definition, upper and lower approximative sums
12 Definition, upper and lower integrals
13 Definition, integrable function
14 Theorem, continuous functions are integrable
15 Properties of the integral
16 The fundamental theorem of calculus
17 Theorem, evaluation of the definite integral
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