The Fundamental Theorem of Calculus
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Theorem, evaluation of the definite integral
Equations are omitted for technical reasons - download the original pdf
Let f be integrable in the closed interval [Equation] and let [Equation]be a function such that [Equation] then [Equation] Proof Consider the function [Equation]. Then [Equation] Then by the mean value theorem[Equation] is a constant function. Hence [Equation] Now [Equation] Hence [Equation] Substituting (1) and (2) above gives [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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