The Fundamental Theorem of Calculus
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Theorem, continuous functions are integrable
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Proof Let f be a continuous function in the interval [Equation]. Then, for any , there exists a such that for any two points in [Equation] such that [Equation], then [Equation] [This theorem follows from the definition of continuity.] It follows from that that if D is any dissection with norm less than [Equation], then for any subinterval of this dissection with [Equation] on the rth interval, [Equation] that [Equation] Then [Equation] Theorem, monotonic functions are integrable Proof Let f be a monotonic function on the closed interval [Equation]; then it is either increasing or decreasing. Let us suppose that it is increasing. Let the rth interval be [Equation] ; then, for any x in this interval [Equation]. Furthermore, [Equation]. Then for any dissection D [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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