In this thesis we explore the axiomatic approach to the Riemann integral. We will establish that integration can be formalized in a more axiomatic manner, allowing for a systematic determination of lengths, areas, and volumes, defined by graphs of functions, without using Riemann sums. The focus is on two key theorems that enable this theory. Then we extend the analysis and introduce a third theorem for evaluating integrals in the three-dimensional space.
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