The object of this paper is to define, to establish necessary and sufficient conditions for the existence of, and to consider the elementary properties of the Riemann definite integral of a bounded function.

The main goal of my diploma work on the topic "The Riemann integral and its applications" is to create an overview that describes the individual use of the Riemann integral. The first three chapters deal with the history, the definition of the Riemann integral and with the calculation of integration. There is the overview of the individual use in the fourth chapter (area plane region, length plane curve, volume solid of revolution and area lateral surface solid of revolution) that is accompanied by solved examples, the examples are accompanied by graphical representation for a better idea. In the fifth chapter there is a collection of unsolved examples, for practising of the individual use. The collection is complemented by results for the reader to check the accuracy of calculation.