The Fundamental Theorem of Calculus
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Integration as the reverse of differentiation
Equations are omitted for technical reasons - download the original pdf
The Fundamental Theorem of Calculus states that the process of integration is the reverse of the process of differentiation. If a function [Equation goes here - download the original to see it.] is the derivative of another function [Equation] then [Equation] is the integral of [Equation]. This means[Equation] and [Equation] where [Equation] is the derivative of [Equation]. Diagrammatically, this is [Equation] The purpose of this chapter is to rigorously prove this result. To do so, we need results about limits.
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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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