Abstract
We present in this paper several examples of Lebesgue integral calculated directly from its definitions using Mathematica. Calculation of Riemann integrals directly from its definitions for some elementary functions is standard in higher mathematics education. But it is difficult to find analogical examples for Lebesgue integral in the available literature. The paper contains Mathematica codes which we prepared to calculate symbolically Lebesgue sums and limits of sums. We also visualize the graphs of simple functions used for approximation of the integrals. We also show how to calculate the needed sums and limits by hand (without CAS). We compare our calculations in Mathematica with calculations in some other CAS programs such as wxMaxima, MuPAD and Sage for the same integrals.
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References
Aliprantis, C.D., Burkinshaw, O.: Principles of Real Analysis, 2nd edn. Academic Press, Cambridge (1991)
Antonov, A.: Mathematica presentation of Lebesgue integration. http://demonstrations.wolfram.com/LebesgueIntegration/
Apostol, T.M.: Calculus, Volume 1, One-Variable Calculus with an Introduction to Linear Algebra, 2nd edn. Addison-Wesley Publishing Company, Boston (1991)
Bartle, R.G.: The Elements of Integration and Lebesgue Measure. Wiley, Hoboken (1995)
Benedetto, J.J., Czaja, W.: Integration and Modern Analysis. Birkhäuser, Boston, MA (2009)
Browder, A.: Mathematical Analysis an Introduction, 2nd edn. Springer, Berlin (2001)
Fichtenholz, G.M.: Differential and Integral Calculus, 3rd edn. Fizmatgiz, Moscow (1958)
Folland, G.B.: Real Analysis Modern Technique, 2nd edn. Wiley, Hoboken (2007)
Freniche, F.J.: Mathematica demonstration of Riemann versus Lebesgue integration. http://demonstrations.wolfram.com/RiemannVersusLebesgue/
Jones, F.: Lebesgue Integration on Euclidean Space. Jones & Bartlett Learning, Burlington (2000)
Kołodziej, W.: Mathematical Analysis. Polish Scientific Publishers PWN, Warsaw (2012). (in Polish)
Larson, R.E., Hostetler, R.P., Edwards, B.H.: Calculus, 6th edn. Houghton Mifflin Company, Boston (1998)
Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill Education, New York City (1973)
Ruskeepaa, H.: Mathematica Navigator: Graphics and Methods of applied Mathematics. Academic Press, Boston (2005)
Sikorski, R.: Differential and Integral Calculus. Functions of Several Variables, 2nd edn. Polish Scientific Publishers PWN, Warsaw (1977). (in Polish)
Wolfram, S.: The Mathematica Book. Wolfram Media Cambridge University Press, Champaign (1996)
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Wojas, W., Krupa, J. Familiarizing Students with Definition of Lebesgue Integral: Examples of Calculation Directly from Its Definition Using Mathematica. Math.Comput.Sci. 11, 363–381 (2017). https://doi.org/10.1007/s11786-017-0321-5
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DOI: https://doi.org/10.1007/s11786-017-0321-5