In the first chapter of this thesis we present the development of method exhaustion, which started some time before the Hellenistic period in ancient Greece. Antiphon first attempted to solve the problem of the quadrature of the circle by inscribing regular polygons in circle. The idea was formally described by Eudoxus of Cnidos in a proposition (known as lemma of Eudoxus) which can be found the X. book of Euclid’s Elements. In the XII. book Euclid gives results obtained by application of this lemma to geometry. The most important contribution to this method was made by Archimedes of Syracuse, who used it to determine areas and volumes of curved plane and solid figures. After Archimedes there was no significant progress in development of this method up until the 17th century. In the 17th century, the method of exhaustion transformed into the modern integral calculus, whose founders are Isaac Newton and Gottfried Wilhelm Leibniz. In the last chapter we use on the example of area determination to give a short didactical comment on the application exhaustion approach in teaching of mathematics.