A Garden of Integrals

A Garden of Integrals

Frank E. Burk
Volume: 31
Copyright Date: 2007
Edition: 1
Pages: 296
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  • Book Info
    A Garden of Integrals
    Book Description:

    The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reason for their existence and their uses are given. Historical information is plentiful. Advanced undergraduate mathematics majors, graduate students, and faculty members are the audience for the book. Even experienced faculty members are unlikely to be aware of all of the integrals in the Garden of Integrals and the book provides an opportunity to see them and appreciate the richness of the idea of integral. Professor Burke's clear and well-motivated exposition makes this book a joy to read.

    eISBN: 978-1-61444-209-7
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Foreword
    (pp. ix-x)

    From quadratures of lunes to quantum mechanics, the development of integration has a long and distinguished history. Chapter 1 begins our exploration by surveying some of the historical highlights—milestones in man’s capacity to think rationally. Whether motivated by applied considerations (areas, heat flow, particles in suspension) or aesthetic results such as


    the common thread has been and will continue to beunderstanding.

    Mathematical discoveries are but markers in our quest to understand our place in this universe. In the profession of mathematics, we are all too frequently humbled, but we persevere for those rare moments of exhilaration that...

  3. Table of Contents
    (pp. xi-xiv)
  4. CHAPTER 1 An Historical Overview
    (pp. 1-28)

    Figures 1(a) through (d) demonstrate the general idea of rearranging a given area to form another shape. In the first example, we have a circle rearranged into a parallelogram by a method that has been known for hundreds of years. Figure 1(e) represents a different manipulation of area calledscalingwhere, despite enlargement or shrinkage, shape and proportion are retained.

    Hippocrates (430 b.c.e.), a merchant of Athens, was one of the first to find the area of a plane figure (lune) bounded by curves (circular arcs). The crescent-shaped region whose area is to be determined is shown in Figure 2....

  5. CHAPTER 2 The Cauchy Integral
    (pp. 29-44)

    Augustin-Louis Cauchy (1789–1857) was the founder of integration theory. Before Cauchy, the emphasis was on calculating integrals of specific functions. For example, in our calculus courses we use the formulae

    1+2+\cdots +n=\frac{n(n+1)}{2}and

    1^{2}+2^{2}+\cdots +n^{2}=\frac{n(n+1)(2n+1)}{6}

    to show, using approximation by interior and exterior rectangles, that the areas of the regions between thex-axis and the curvesy=xandy=x² for0\leq x\leq bare given by\int _{0}^{b}x\; dx=b^{2}/2and\int _{0}^{b}x^{2}\; dx=b^{3}/3— results obtained much earlier by Archimedes (287–212 b.c.e).

    Such beautiful results begged for extension, most accomplished by sheer ingenuity. Here are some examples, formulae owing to Fermat, Wallis,...

  6. CHAPTER 3 The Riemann Integral
    (pp. 45-74)

    The Riemann integral (1854)—the familiar integral of calculus developed by Bernhard Riemann (1826–1866)—was a response to various questions raised by Dirichlet about just how discontinuous a function could be and still have a well-defined integral.

    Given a bounded functionfon the interval [a,b], divide [a,b] into a finite number of contiguous subintervals[x_{k-1},\: x_{k}], witha=x0<x1<···<xn=b, and pick a pointc_{k}in[x_{k-1},\: x_{k}]. As with Cauchy integration, we begin by defining some terminology.

    The collection of point intervals

    (c_{1},[x_{0},x_{1}]),\; (c_{2},[x_{1},x_{2}]),\ldots,(c_{n},[x_{n-1},x_{n}])

    is called aRiemann partitionof [a,b],...

  7. CHAPTER 4 The Riemann–Stieltjes Integral
    (pp. 75-84)

    After Riemann’s formulation of the integral, various generalizations were attempted. One of the most successful, the so-calledRiemann–Stieltjes integral, was obtained by Thomas Stieltjes (1856–1894). Stieltjes was trying to model mathematically the physical problem of computing moments for various mass distributions on thex-axis, with massesm_{i}at distancesd_{i}from the origin (Birkhoff, 1973):

    “Such a distribution will be perfectly determined if one knows how to calculate the total mass distributed over each segmentOx[ofOX]. This evidently will be an increasing function ofx.... Accordingly, letϕ(x) be an increasing function defined on the interval...

  8. CHAPTER 5 Lebesgue Measure
    (pp. 85-110)

    Vito Volterra’s 1881 example of a function having a bounded derivative that was not Riemann integrable (Section 3.12) prompted Henri Lebesgue to develop a method of integration to remedy this shortcoming: The Lebesgue integral of a bounded derivative returns the original function.

    Lebesgue’s integration process was fundamentally different from that of his predecessors. His simple but brilliant idea:Partition the range of the function rather than its domain.

    How does this work? Suppose the functionfis bounded on the interval [a,b], sayα<f<β. Partition the interval [α,β],

    α=y0<y1<y2< ··· <yn-1<yn=β,

    with (math) denoting...

  9. CHAPTER 6 The Lebesgue Integral
    (pp. 111-154)

    The culmination of our efforts regarding measure theory, Lebesgue integration, is a mathematical idea with numerous and far-reaching applications. We will confine our remarks to the essential concepts, but the interested reader will be well rewarded by additional efforts.

    We begin our exploration of Lebesgue’s integral (1902) by defining what it means to beLebesgue integrable.

    Supposefis a bounded measurable function on the interval [a,b], soα<f<β. Partition the range off:α=y0<y1< ··· <yn=β, and letEk= {x∈ [a,b]|yk-1f<yk}, fork= 1,2,...,n.

    Form the lower sum\sum _{k=1}^{n} y_{k-1}\mu (Ek)...

  10. CHAPTER 7 The Lebesgue–Stieltjes Integral
    (pp. 155-168)

    The Lebesgue measure weights intervals according to their length:μ((a,b]) =ba. We are looking for different weightings of intervals, trying to find measures different from Lebesgue measure. Of course the general properties of a measure — nonnegative and countable additivity for disjoint sequences of measurable sets — must be retained. We will discuss three of the most common approaches, which differ from each other essentially in their starting points.

    In Chapter 4 we explored the particularly fruitful, and most fundamental, approach of Thomas Stieltjes, who modelled mass distributions with monotone increasing right-continuous functions. His approach involved...

  11. CHAPTER 8 The Henstock–Kurzweil Integral
    (pp. 169-204)

    In this chapter we present a beautiful extension of the Lebesgue integral obtained by an apparently slight modification of the Riemann integration process. Recall that in Section 3.12 we saw functions with a bounded derivative whose derivative was not Riemann integrable. These examples prompted Lebesgue to develop an integration process by which differentiable functions with bounded derivatives could be reconstructed from their derivatives:

    \mathrm{L}\int _{a}^{x}F'd\mu =F(x)-F(a).

    This was a Fundamental Theorem of Calculus for the Lebesgue integral (Theorem 6.4.2).

    The next step would be to try to remove the “bounded” requirement on the derivative. We want an integration process in which...

  12. CHAPTER 9 The Wiener Integral
    (pp. 205-234)

    In the preceding chapters, the integrals under discussion were defined on sets of real numbers. So the domains of integration have consisted of real numbers. In contrast, the Wiener integral has as its domain of integrationthe space of continuous functions on the interval[0, 1]that begin at the origin. A continuous function now plays the role of a real number.

    With the Wiener integralpathreplacespoint: these are integrals over sets of continuous functions, integrals over “paths.” Hence the terminologypath integral. The approach, as in the development of the Lebesgue integral, is threefold:

    1. We begin by...

  13. CHAPTER 10 The Feynman Integral
    (pp. 235-278)

    In the 1920s Norbert Wiener introduced the concept ofa measure on the space of continuous functions. As you recall from Chapter 9, this idea arose from his attempts to model the Brownian motion of small particles suspended in a fluid. In the 1940s Richard Feynman (1918–1988) developed an integral on the same space of continuous functions in his efforts to model the quantum mechanics of very small particles such as electrons. To succeed, Feynman’spath integralapproach to quantum mechanics had to be consistent with Schrödinger’s Equation.

    A frequent correspondent of Albert Einstein, Erwin Schrödinger (1887–1961) made...

  14. Index
    (pp. 279-280)
  15. About the Author
    (pp. 281-281)