The Fundamental Theorem of Calculus
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Definition, integrable function
Equations are omitted for technical reasons - download the original pdf
Above we gave a definition of a function derived from the norm of a dissection [Equation goes here - download the original to see it.] If [Equation] as [Equation] then the function f is said to be integrable. The integral is written [Equation] Theorem Using the notation of the preceding section [Equation] as [Equation] f is integrable Proof (1) [Equation] as [Equation] f is integrable Suppose [Equation]. Then, Given [Equation] there is a [Equation] such that for any dissection D, [Equation]. We can write [Equation] as a sum of terms as follows [Equation] where [Equation] Hence [Equation] This is true for any value of [Equation]. Hence [Equation]. Hence [Equation] which is bounded by J above and j below also tends to limit 0. (2) f is integrable [Equation] as [Equation] Let f be integrable in [Equation]. Then [Equation] as [Equation] Let [Equation]. Then for all [Equation] there is a [Equation] such that for all dissections D with norm less than [Equation] we have [Equation] As before, let [Equation] on the rth interval, [Equation]. Then [Equation] Hence [Equation] Therefore [Equation]Likewise [Equation] Therefore [Equation] Since this is true for all numbers [Equation]
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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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