Abstract
Calculus has two components, and, thus far, we have been dealing with only one of them, namely differentiation. Differentiation is a systematic procedure for disassembling quantities at an infinitesimal level. Integration, which is the second component and is the topic here, is a systematic procedure for assembling the same sort of quantities. One of Newton’s great discoveries is that these two components complement one another in a way that makes each of them more powerful.
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Notes
- 1.
Although it was Newton who made this result famous, it had antecedents in the work of James Gregory and Newton’s teacher Isaac Barrow. Mathematicians are not always reliable historians, and their attributions should be taken with a grain of salt.
- 2.
In general, a function \(f:{\mathbb R}\longrightarrow \mathbb C\) is said to be periodic if there is some \(\alpha >0\) such that \(f(x+\alpha )=f(x)\) for all \(x\in {\mathbb R}\), in which case \(\alpha \) is said to be a period of f. Here, without further comment, we will always be dealing with the case when \(\alpha =1\) unless some other period is specified.
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© 2015 Springer International Publishing Switzerland
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Stroock, D.W. (2015). Integration. In: A Concise Introduction to Analysis. Springer, Cham. https://doi.org/10.1007/978-3-319-24469-3_3
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DOI: https://doi.org/10.1007/978-3-319-24469-3_3
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