Skip to main content
SpringerLink
Log in

The Riemann Integral

  • Chapter
  • First Online:
Real Analysis on Intervals

  • 1581 Accesses

Abstract

Since antiquity, people were interested in computing the length of curves, the area of surfaces, and the volumes of solids.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bourbaki, N.: Fonctions d’une variable r éelle. Springer, Berlin (2007)

    Google Scholar 

  2. Ritt, J.F.: Integration in Finite Terms: Liouville’s Theory of Elementary Methods. Columbia University Press, New York (1948)

    MATH  Google Scholar 

  3. Adamchik, V., Wagon, S.: A simple formula for \({\pi }\). Am. Math. Mon. 104, 852–855 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Stark, E.: Application of a mean value theorem for integrals to series summation. Am. Math. Mon. 85, 481–483 (1978)

    Article  MATH  Google Scholar 

  5. Lanczos, C.: Applied Analysis. Prentice-Hall, Englewood Cliffs (1956)

    Google Scholar 

  6. Riemann, B.: Ueber die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe. Read in 1854, published in 1867. Republished in Riemann’s Gesammelte Math. Werke, 1892, pp. 227–271. Reprint, Dover Publications Inc, New York (1953)

    Google Scholar 

  7. Niculescu, C.P., Persson, L.-E.: Convex Functions and Their Applications. A Contemporary Approach. CMS Books in Mathematics. Springer, New York (2006)

    MATH  Google Scholar 

  8. Niculescu, C.P.: The Hermite-Hadamard inequality for log-convex functions. Nonlinear Anal. 75, 662–669 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Mitroi, F.C., Niculescu, C.P.: An Extension of Young’s Inequality. Abstract and Applied Analysis, Article ID 162049, vol. 2011, p. 18

    Google Scholar 

  10. Niederreiter, H.: Random number generation and Quasi-Monte Carlo methods. Soc. Ind. Appl. Math. (1992)

    Google Scholar 

  11. Weyl, H.: Ueber die Gleich-verteilung von Zahlen mod. Eins. Math. Ann. 77(3), 313–352 (1916)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Dover Publishing, Mineola (2012)

    Google Scholar 

  13. Diaconis, P.: The distribution of leading digits and uniform distribution mod 1. Ann. Probab. 5(1), 72–81 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  14. Tolstov, G.P.: Parametric differentiation, and the restricted Denjoy integral. Mat. Sb. 53, 387–392 (1961) (Russian)

    Google Scholar 

  15. Bartle, R.G.: A Modern Theory of Integration. Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  16. Gordon, R.A.: The Integrals of Lebesgue, Denjoy, Perron and Henstock. Graduate Studies in Mathematics. American Mathematical Society, Providence (1994)

    MATH  Google Scholar 

  17. Lee, P.Y., Výborný, R.: The Integral: An Easy Approach After Kurzweil and Henstock. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  18. Bartle, R.G.: Return to the Riemann integral. Am. Math. Mon. 103, 625–632 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  19. Cousin, P.: Sur les fonctions de n variables complexes. Acta. Math. 19, 1–62 (1895)

    Google Scholar 

  20. Dunham, W.: The Calculus Gallery: Masterpieces from Newton to Lebesgue. Princeton University Press, Princeton (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Abdus Salam School of Mathematical Sciences, Lahore, Pakistan

    A. D. R. Choudary

  2. Department of Mathematics, University of Craiova, Craiova, Romania

    Constantin P. Niculescu

Authors
  1. A. D. R. Choudary
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Constantin P. Niculescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. D. R. Choudary .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer India

About this chapter

Cite this chapter

Choudary, A.D.R., Niculescu, C.P. (2014). The Riemann Integral. In: Real Analysis on Intervals. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2148-7_9

Download citation

Publish with us

Policies and ethics

Search

Navigation