The Fundamental Theorem of Calculus
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Fundamental Theorem of Calculus
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The purpose of this chapter is to establish a rigorous foundation for the integral calculus. It is not normal when introducing the integral calculus for the first time to teach it from first principles. Therefore, it is in fact assumed here that the student is already familiar with the integral calculus and its applications. The rigorous foundation of the calculus requires knowledge of limits. It is assumed that the student by this stage has already encountered the definition of limits, building on the theory of sequences and series. The student is also expected to be familiar with the definition of a continuous function and with the properties of differentiable functions, particularly the mean value theorem.
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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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