In this thesis, We tried to introduce definite integration to the curriculum of high school mathematics with numerical integration, which had been introduced with quadrature method. For this purpose, We used new experimental mathematics approaches, so-called investigation and examination. In chapter II, We examined how much computers had been used in teaching mathematics. In chapter III, We presented the theoretical background of approximation integration within numerical integration. In chapter IV, We studied and compared various methods of numerical integration, and examined the relation between curvature of a curved line and numerical integration. In order to study more easily, We used some of computer programs. We hope that this thesis will be a turning point in developing new teaching methods and improving curriculum of mathematics in high school.