THE CALCULUS INTEGRAL Brian S. Thomson Simon Fraser University CLASSICALREALANALYSIS.COM This text is intended as an outline for a rigorous course introducing the basic ele- ments of integration theory to honors calculus students or for an undergraduate course in elementary real analysis. Since “all” exercises are worked through in the appendix, the text is particularly well suited to self-study. For further information on this title and others in the series visit our website. www.classicalrealanalysis.com There are PDF files of all of our texts available for download as well as instructions on how to order trade paperback copies. We always allow access to the full content of our books on GOOGLE BOOKS and on the AMAZON Search Inside the Book feature. Cover Image: Sir Isaac Newton And from my pillow, looking forth by light Of moon or favouring stars, I could behold The antechapel where the statue stood Of Newton with his prism and silent face, The marble index of a mind for ever Voyaging through strange seas of Thought, alone. William Wordsworth, The Prelude. Citation: The Calculus Integral, Brian S. Thomson, ClassicalRealAnalysis.com (2010), [ISBN 1442180951] Date PDF file compiled: June 19, 2011 BETA VERSION β1.0 The file or paperback that you are reading should be considered a work in progress. In a classroom setting make sure all participants are using the same beta version. We will add and amend, depending on feedback from our users, until the text appears to be in a stable condition. ISBN: 1442180951 EAN-13: 9781442180956 CLASSICALREALANALYSIS.COM PREFACE There are plenty of calculus books available, many free or at least cheap, that discuss integrals. Why add another one? Our purpose is to present integration theory at an honors calculus level and in an easier manner by defining the definite integral in a very traditional way, but a way that avoids the equally traditional Riemann sums definition. Riemann sums enter the picture, to be sure, but the integral is defined in the way that Newton himself would surely endorse. Thus the fundamental theorem of the calculus starts off as the definition and the relation with Riemann sums becomes a theorem (not the definition of the definite integral as has, most unfortunately, been the case for many years). As usual in mathematical presentations we all end up in the same place. It is just that we have taken a different route to get there. It is only a pedagogical issue of which route offers the clearest perspective. The common route of starting with the definition of the Riemann integral, providing the then necessary detour into improper integrals, and ultimately heading towards the Lebesgue integral is arguably not the best path although it has at least the merit of historical fidelity. Acknowledgments I have used without comment material that has appeared in the textbook [TBB] Elementary Real Analysis, 2nd Edition, B. S. Thomson, J. B. Bruck- ner, A. M. Bruckner, ClassicalRealAnalyis.com (2008). I wish to express my thanks to my co-authors for permission to recycle that material into the idiosyncratic form that appears here and their encouragement (or at least lack of discouragement) in this project. I would also like to thank the following individuals who have offered feedback on the material, or who have supplied interesting exercises or solutions to our exercises: [your name here], ... i ii Note to the instructor Since it is possible that some brave mathematicians will undertake to present integra- tion theory to undergraduates students using the presentation in this text, it would be appropriate for us to address some comments to them. What should I teach the weak calculus students? Let me dispense with this question first. Don’t teach them this material, which is aimed much more at the level of an honor’s calculus course. I also wouldn’t teach them the Riemann integral. I think a reasonable outline for these students would be this: 1. An informal account of the indefinite integral formula F (x)dx = F(x)+C Z ′ just as an antiderivative notation with a justification provided by the mean-value theorem. 2. An account of what it means for a function to be continuous on an interval [a,b]. 3. The definition b F′(x)dx = F(b) F(a) Za − for continuous functions F : [a,b] R that are differentiable at all1 points in → (a,b). The mean-value theorem again justifies the definition. You won’t need improper integrals, e.g., 1 1 1 d dx = 2√x dx = 2 0. Z0 √x Z0 dx − 4. Any properties of integrals that are direct translations of derivative properties. 5. The Riemann sums identity b n f (x)dx = ∑ f (ξ )(x x ) Z i∗ i i 1 a i=1 − − ξ where the points i∗ that make this precise are selected by the mean-value theo- rem. 1. or all but finitely many points iii iv 6. The Riemann sums approximation b n f (x)dx ∑ f (ξ )(x x ) Z i i i 1 a ≈ i=1 − − where the points ξi can be freely selected inside the interval. Continuity of f ξ ξ justifies this since f ( i) f ( i∗) if the points xi and xi 1 are close together. [It ≈ − is assumed that any application of this approximation would be restricted to con- tinuous functions.] That’s all! No other elements of theory would be essential and the students can then focus largely on the standard calculus problems. Integration theory, presented in this skeletal form, is much less mysterious than any account of the Riemann integral would be. On the other hand, for students that are not considered marginal, the presentation in the text should lead to a full theory of integration on the real line provided at first that the student is sophisticated enough to handle ε, δ arguments and simple compactness proofs (notably Bolzano-Weierstrass and Cousin lemma proofs). Why the calculus integral? Perhaps the correct question is “Why not the Lebesgue integral?” After all, integration theory on the real line is not adequately described by either the calculus integral or the Riemann integral. The answer that we all seem to have agreed upon is that Lebesgue’s theory is too difficult for beginning students of integration theory. Thus we need a “teaching inte- gral,” one that will present all the usual rudiments of the theory in way that prepares the student for the later introduction of measure and integration. Using the Riemann integral as a teaching integral requires starting with summations and a difficult and awkward limit formulation. Eventually one reaches the fundamental theorem of the calculus. The fastest and most efficient way of teaching integration theory on the real line is, instead, at the outset to interpret the calculus integral b F′(x)dx = F(b) F(a) Za − as a definition. The primary tool is the very familiar mean-value theorem. That theorem leads quickly back to Riemann sums in any case. The instructor must then drop the habit of calling this the fundamental theorem of the calculus. Within a few lectures the main properties of integrals are available and all of the computational exercises are accessible. This is because everything is merely an immediate application of differentiation theorems. There is no need for an “improper” theory of the integral since integration of unbounded functions requires no additional ideas or lectures. There is a long and distinguished historical precedent for this kind of definition. For all of the 18th century the integral was understood only in this sense2 The descriptive definition of the Lebesgue integral, which too can be taken as a starting point, is exactly 2Certainly Newton and his followers saw it in this sense. For Leibnitz and his advocates the integral was a sum of infinitesimals, but that only explained the connection with the derivative. For a lucid account of the thinking of the mathematicians to whom we owe all this theory see Judith V. Grabiner, Who gave v the same: but now requires F to be absolutely continuous and F′ is defined only almost everywhere. The Denjoy-Perron integral has the same descriptive definition but relaxes the condition on F to that of generalized absolute continuity. Thus the narrative of integration theory on the real line can told simply as an interpretation of the integral as meaning merely b F′(x)dx = F(b) F(a). Za − Why not the Riemann integral? Or you may prefer to persist in teaching to your calculus students the Riemann integral and its ugly step-sister, the improper Riemann integral. There are many reasons for ceasing to use this as a teaching integral; the web page, “Top ten reasons for dumping the Riemann integral” which you can find on our site www.classicalrealanalysis.com has a tongue-in-cheek account of some of these. The Riemann integral does not do a particularly good job of introducing integration theory to students. That is not to say that students should be sheltered from the notion of Riemann sums. It is just that a whole course confined to the Riemann integral wastes considerable time on a topic and on methods that are not worthy of such devotion. In this presentation the Riemann sums approximation to integrals enters into the discussion naturally by way of the mean-value theorem of the differential calculus. It does not require several lectures on approximations of areas and other motivating stories. The calculus integral For all of the 18th century and a good bit of the 19th century integration theory, as we understand it, was simply the subject of antidifferentiation.