Abstract
Here we consider the familiar integral from calculus which is generally attributed to Riemann, though the idea of upper and lower sums for finding areas was used previously by Cauchy; other mathematicians had used such sums before Cauchy for estimating integrals but not for calculating exact values. The method we present is cleaner from the technical point of view and is attributed to Darboux; this method employs the notions of upper and lower integrals. To avoid confusion we will discuss the equivalence of this approach with the familiar Riemann sum approach. We will also devote some time to the famed Fundamental Theorem of Calculus and use this to derive some of the familiar properties of the exponential and logarithmic functions. The final section in this chapter will investigate the dysfunctional relationship between the limit process (with regard to a sequence of functions) and the Riemann integral propelling us toward the more sturdy Lebesgue integral.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abbott, S.: Understanding Analysis. Springer, New York (2001)
Axler, S.: Linear Algebra Done Right. Springer, New York (1996)
Axler, S.: Down with determinants! Am. Math. Mon. 102, 139–154 (1995)
Bartle, R.G.: Return to the Riemann integral. Am. Math. Mon. 103, 625–632 (1996)
Beanland, K., Roberts, J.W., Stevenson, C.: Modifications of Thomae’s function and differentiability. Am. Math. Mon. 116(6), 531–535 (2009)
Beauzamy, B.: Introduction to Operator Theory and Invariant Subspaces. North-Holland, Amsterdam (1988)
Bressoud, D.: A Radical Approach to Lebesgue’s Theory of Integration. MAA Textbooks. Cambridge University Press, Cambridge (2008)
Bressoud, D.: A Radical Approach to Real Analysis, 2nd edn. Mathematical Association of America, Washington (2007)
de Silva, N.: A concise, elementary proof of Arzelà’s bounded convergence theorem. Am. Math. Mon. 117(10), 918–920 (2010)
Darst, R.B.: Some Cantor sets and Cantor functions. Math. Mag. 45, 2–7 (1972)
Drago, G., Lamberti, P.D., Toni, P.: A “bouquet” of discontinuous functions for beginners of mathematical analysis. Am. Math. Mon. 118(9), 799–811 (2011)
Halmos, P.: A Hilbert Space Problem Book. Springer, New York (1982)
Harier, E., Wanner, G.: Analysis by Its History. Springer, New York (1996)
Kirkwood, J.R.: An Introduction to Analysis. Waveland, Prospect Heights (1995)
Knopp, K.: Theory and Application of Infinite Series. Blackie, London (1951)
Kubrusly, C.: Elements of Operator Theory. Birkhäuser, Boston (2001)
Lebesgue, H.: Measure and the Integral. Holden-Day, San Francisco (1966)
MacCluer, B.D.: Elementary Functional Analysis. Springer, New York (2009)
Martinez-Avendano, R., Rosenthal, P.: An Introduction to Operators on the Hardy-Hilbert Space. Springer, New York (2006)
Mihaila, I.: The rationals of the Cantor set. Coll. Math. J. 35(4), 251–255 (2004)
Nadler, S.B., Jr.: A proof of Darboux’s theorem. Am. Math. Mon. 117(2), 174–175 (2010)
Niven, I.: Numbers: Rational and Irrational. New Mathematical Library, vol. 1. Random House, New York (1961)
Niven, I.: Irrational Numbers. The Carus Mathematical Monographs, vol. 11. Mathematical Association of America, New York (1956)
Olmsted, J.M.H.: Real Variables. Appleton-Century-Crofts, New York (1959)
Patty, C.W.: Foundations of Topology. Waveland, Prospect Heights (1997)
Pérez, D., Quintana, Y.: A survey on the Weierstrass approximation theorem (English, Spanish summary). Divulg. Mat. 16(1), 231–247 (2008)
Radjavi, H., Rosenthal, P.: The invariant subspace problem. Math. Intell. 4(1), 33–37 (1982)
Rynne, B.P., Youngson, M.A.: Linear Functional Analysis. Springer, London (2008)
Saxe, K.: Beginning Functional Analysis. Springer, New York (2002)
Strichartz, R.S.: The Way of Analysis. Jones and Bartlett, Boston (1995)
Thompson, B.: Monotone convergence theorem for the Riemann integral. Am. Math. Mon. 117(6), 547–550 (2010)
Tucker, T.: Rethinking rigor in calculus: the role of the mean value theorem. Am. Math. Mon. 104(3), 231–240 (1997)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Pons, M.A. (2014). The Riemann Integral. In: Real Analysis for the Undergraduate. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9638-0_7
Download citation
DOI: https://doi.org/10.1007/978-1-4614-9638-0_7
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-9637-3
Online ISBN: 978-1-4614-9638-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)