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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.

Familiarizing Students with Definition of Lebesgue Integral: Examples of Calculation Directly from Its Definition Using Mathematica