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Although, as will become apparent very soon, integration theory may be developed without any appeal to geometric intuition, one of the primary motivations for dealing with integrals is their connection with the definition and evaluation of areas of certain plane sets; see the comments in VIII.2.4. (We shall nowhere be concerned with the technically more difficult problems concerning areas of portions of curved surfaces.) Because of this, it is as well to preface the detailed treatment of integrals with some comments on the concept of area and how it is often used to bolster (and not merely motivate) the development of integration theory. The customary mixing of the two concepts (area and integral) can generate damaging confusion, to avoid which some conscious effort is necessary.
KeywordsBounded Interval Area Function Filter Base Extended Concept Proper Integral
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