Abstract
This introductory chapter opens with a review of the area interpretation of the Riemann integral, followed with a brief comparison with the Lebesgue integral. The idea of evaluating definite integrals with tricks is introduced and illustrated with some detailed examples. The need to be ever-vigilant for singularities is emphasized and illustrated with an integral made famous in physics by Richard Feynman. Other specific integrals of historical interest are then treated (Dazell’s integral that computes upper and lower bounds on pi, an integral inspired by the physics of Brownian motion which Einstein studied, an integral that results from a nineteenth century exam problem posed by Lord Rayleigh, and an integral developed by Feynman in a probabilistic attempt to ‘solve’ Fermat’s Last Theorem). The Cauchy-Schwarz inequality is derived and then used to study the autocorrelation integral of a real-valued function. The usefulness of employing a computer to study integrals is asserted and then illustrated with examples of the Monte Carlo method for estimating definite integrals.
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Notes
- 1.
You can find high school level proofs of Cantor’s results in my book The Logician and the Engineer, Princeton 2012, pp. 168–172.
- 2.
For more on r(x), see E. Hairer and G. Wanner, Analysis by Its History, Springer 1996, p. 232, and William Dunham, The Calculus Gallery, Princeton 2005, pp. 108–112.
- 3.
Comment made after the presentation by E. C. Francis of his paper “Modern Theories of Integration,” The Mathematical Gazette, March 1926, pp. 72–77.
- 4.
In Hamming’s paper “Mathematics On a Distant Planet,” The American Mathematical Monthly, August–September 1998, pp. 640–650.
- 5.
See Bartle’s award-winning paper “Return to the Riemann Integral,” The American Mathematical Monthly, October 1996, pp. 625–632. He was professor of mathematics at the University of Illinois for many years, and then at Eastern Michigan University.
- 6.
Writing sin(1) means, of course, the sine of 1 radian = \( \frac{180^{{}^{\circ}}}{\uppi}={57.3}^{{}^{\circ}} \) (not of 1°).
- 7.
I am assuming that when you see \( \int \frac{1}{{\mathrm{a}}^2+{\mathrm{x}}^2}\ \mathrm{dx} \) you immediately recognize it as \( \frac{1}{\mathrm{a}}{\tan}^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right) \). This is one of the few ‘fundamental’ indefinite integrals I’m going to assume you’ve seen previously from a first course of calculus. Others are: \( \int \frac{1}{\mathrm{x}}\mathrm{dx}=\ln \left(\mathrm{x}\right) \), ∫ex dx = ex, \( \int {\mathrm{x}}^{\mathrm{n}}\ \mathrm{dx}=\frac{{\mathrm{x}}^{\mathrm{n}+1}}{\mathrm{n}+1}\ \left(\mathrm{n}\ne -1\right) \), \( \int \frac{\mathrm{dx}}{\sqrt{{\mathrm{a}}^2-{\mathrm{x}}^2}}={\sin}^{-1}\left(\frac{\mathrm{x}}{a}\right), \) and ∫ ln (x)dx = xln(x) − x.
- 8.
This result says, if we expand the integrand as a power series,
\( \frac{\uppi}{4}={\int}_0^1\left(1-{\mathrm{x}}^2+{\mathrm{x}}^4-{\mathrm{x}}^6+\dots \right)\mathrm{dx}=\left(\mathrm{x}-\frac{1}{3}{\mathrm{x}}^3+\frac{1}{5}{\mathrm{x}}^5-\frac{1}{7}{\mathrm{x}}^7+\dots \right){\vert}_0^1 \), or
\( \frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dots \), a famous formula discovered (by other means) decades earlier by Leibniz (see note 5 in the original Preface).
- 9.
This integral appeared in Feynman’s famous paper “Space-Time Approach to Quantum Electrodynamics,” Physical Review, September 15, 1949, pp. 769–789. Some historical discussion of the integral is in my book Number–Crunching, Princeton 2011, pp. xx–xxi.
- 10.
‘At random’ has the following meaning. If we look at any tiny patch of area dA in the interior of C1, a patch of any shape, then the probability a point is selected from that area patch is dA divided by the area of C1. We say that each of the three points is selected uniformly from the interior of C1.
- 11.
I think it almost intuitively obvious that the probability is scale-invariant (the same for any value of a), but just in case it isn’t obvious for you I’ll carry the radius of C1 along explicitly. At the end of our analysis you’ll see that the scale-setting parameter a has disappeared, proving my claim.
- 12.
The analysis I’ve just taken you through is the one given on pp. 817–818 of Edwards’ book that I mentioned in the original Preface.
- 13.
This interval of estimates is a result of the code’s use of a random number generator (with the rand command) —every time we run the code we get a new estimate that is (slightly) different from the estimates produced by previous runs.
- 14.
The Educational Times was a self-described “monthly journal of education, science and literature,” published by the College of Preceptors in London, to deal with general educational questions at the school and university level. In the years it appeared, 1847–1918, over 18,000 questions on mathematics were presented and solved.
- 15.
You can find Seitz’s solution reprinted in Problems and Solutions from The Mathematical Visitor 1877–1896 (Stanley Rabinowitz, editor), MathPro Press 1996, pp. 125–126.
- 16.
Imagine that we have defined the maximum absolute value of a step, which will be our unit distance. Then, in one dimension (call it x) the particle moves, after each molecular hit, a distance randomly selected from the interval −1 to +1. In a second, perpendicular direction (call it y) the particle moves, after each molecular hit, a distance randomly selected from the interval −1 to +1. Figure 1.8.2 shows the combined result of these two independent motions for four particles, each for 500 hits (the four curves are MATLAB simulations).
- 17.
If you are curious about the details of such a derivation, you can find them in my book Mrs. Perkins’s Electric Quilt, Princeton University Press 2009, pp. 263–267.
- 18.
For a proof of this, see Mrs. Perkins’s, pp. 282–283.
- 19.
Paul J. Nahin, “Rayleigh’s Rotating Ring,” The Mathematical Intelligencer, Summer 2017, pp. 72–75. The answer to Rayleigh’s question is that the ring reaches its structural limit at just a bit <2700 rpm. This problem isn’t as theoretical as it might appear. A specific concern over high-speed rotational disintegration occurred when Rayleigh’s friend, the English physicist Oliver Lodge (1851–1940), conducted experiments in a study of the ether (the mythical substance Victorian physicists at one time believed filled all of what appeared to be empty space, through which electromagnetic waves, light, could travel). Lodge’s experimental setup involved the high-speed rotation of massive steel plates and, by Christmas of 1891, he was operating them at 2800 rpm. At that speed, one of Lodge’s friends worried that if there were any flaws in the plates they could disintegrate, and as Lodge himself wrote in a lab notebook, “we should have our heads cut off.”
- 20.
Named after the French lawyer Pierre de Fermat (1601–1665), who sometime around 1637 wrote in the margin of one of his private books that he had discovered a truly wonderful proof that there are no integer solutions to xn + yn = zn for any n ≥ 3. As is well-known, Fermat didn’t provide that proof, claiming the margin too small to hold it. It wasn’t until more than three centuries later that a proof was finally found (mathematicians believe, today, that whatever Fermat thought he had in 1637, he later realized was flawed).
- 21.
I’ve taken the Feynman quotes from Silvan S. Schweber, QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga, Princeton 1994, p. 464. ‘QED’ is quantum electrodynamics, the theory that won Feynman his share of the 1965 Nobel Prize in physics.
- 22.
In Ralph Palmer Agnew’s 1960 Differential Equations we read the following words (page 370): “This [the Cauchy-Schwarz inequality] is an exceptionally potent weapon. There are many occasions on which persons who know and think of using this formula can shine while their less fortunate brethren flounder.”
- 23.
This way of generating random locations over a circle is called the rejection method . It’s not computationally efficient, but does have the virtue of being intuitive. A more sophisticated (but less intuitive) way to directly generate uniformly distributed points over a circular region is discussed in my book Digital Dice, Princeton 2008, pp. 16–18.
- 24.
Don’t continue reading until you fully understand the limits on the two integrals. Since the second circle in Fig. 1.8.5 has radius 1, and is centered on x = s, then that circle crosses the x-axis at s − 1 (which is, of course, non-positive as s is, at most, 1). Also, since both circles have radius 1, the isosocles triangle in Fig. 1.8.5 shows that the two circles intersect at \( \mathrm{x}=\frac{1}{2}\mathrm{s} \).
- 25.
In the essay titled “A Different Box of Tools,” in Surely You’re Joking, Mr. Feynman!, W. W. Norton 1985, pp. 84–87.
- 26.
I sat for the Putnam in 1959, and I’m pretty sure I know on which side of the median score I fell!
- 27.
In the essay “Los Alamos from Below” in Surely You’re Joking, Mr. Feynman!, pp. 107–136.
- 28.
D. S. Mitrinović and J. D. Kečkić, The Cauchy Method of Residues, D. Reidel 1984 (originally published in 1978, in Serbian), pp. 191–192. This outstanding book is, alas, out-of-print and available today only through used book dealers on the Web (at outrageous prices). Your best bet is to check the holdings of a local college library.
- 29.
For a discussion of how this integral appears in a physics problem, see my book Mrs. Perkins’s Electric Quilt, Princeton 2009, pp. 2–3 (and also that book’s p. 4, for how to attack the challenge question—but try on your own before looking there or at the solutions).
- 30.
The G. H. Hardy Reader, Cambridge University Press 2015, p. 228.
- 31.
J. H. Bartlett, Jr., “The Helium Wave Equation,” Physical Review, April 15, 1937, pp. 661–669.
- 32.
Leo Lavatelli, “The Resistive Net and Finite-Difference Equations,” American Journal of Physics, September 1972, pp. 1246–1257. I’ll tell you more about this famous circuit (involving a spectacular double integral) in the next chapter.
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Nahin, P.J. (2020). Introduction. In: Inside Interesting Integrals. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-43788-6_1
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