The Fundamental Theorem of Calculus
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Properties of the integral
Equations are omitted for technical reasons - download the original pdf
Suppose we have [Equation goes here - download the original to see it.]; it follows from the definition of the integral that [Equation] The following properties of the integral can be established. (1) Let f be integrable in the closed interval [Equation]. Suppose [Equation]. Then f is integrable in [Equation]. (2) Let f be integrable in the closed interval [Equation]. Given [Equation] then [Equation] (3) Let f be integrable in the closed interval [Equation]. For any constant k [Equation] (4) Let f and g be integrable in the closed interval [Equation]. Then their sum is integrable, and [Equation] (5) Let f be integrable in the closed interval [Equation]. Suppose that for all x in [Equation] [Equation] then [Equation] (6) Let f and g be integrable in the closed interval [Equation]. Then their product fg is integrable in [Equation]. If, in addition, we have [Equation] and [Equation] then [Equation] This is a corollary of (5).
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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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