The Riemann Integral

  • Charles H. C. Little
  • Kee L. Teo
  • Bruce van Brunt
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

Riemann integrals of functions are defined and the mean value theorem for integrals is proved. The relationship between integration and differentiation is uncovered in the fundamental theorem of calculus. The techniques of integration by substitution and by parts are established, and the latter is used to develop Stirling’s formula for the approximation of the factorial of a positive integer. It is also used to prove the irrationality of the constants π and e. Integrals are employed to define the concept of arc length of a curve and this idea is then used to give a geometric interpretation of π. Methods for approximating definite integrals are discussed. The chapter concludes with an introduction to improper integrals and the integral test for convergence of a series.

Key words

Riemann integrals Mean value theorem for integrals Fundamental theorem of calculus Integration by substitution Integration by parts Wallis’s formula Stirling’s formula Irrationality of π and e Arc length Geometric interpretation of π Cauchy–Schwarz inequality Numerical integration Improper Integrals Integral test for convergence of a series 

Bibliography

  1. 3.
    Bartle, R.G., Sherbert, D.R.: Introduction to Real Analysis. Wiley, New York (1982)Google Scholar
  2. 7.
    Fulks, W.: Advanced Calculus, 3rd edn. Wiley, New York (1978)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Charles H. C. Little
    • 1
  • Kee L. Teo
    • 1
  • Bruce van Brunt
    • 1
  1. 1.Institute of Fundamental SciencesMassey UniversityPalmerston NorthNew Zealand

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