Subject of this final paper is numerical integration. In order to introduce methods for numerical integration, we will briefly describe trapezoidal rule, Newton-Cotes formulas and Simpson’s rule, also we will state error bound for each method. Main focus of the paper will be Gaussian quadrature. We will explain the key idea underlying this method and derive general form of it. Gauss-Legendre quadrature will be explained in more detail. Applying Peano kernel theory we will obtain error bounds for this method. To sum up, there will be given few examples to compare how each methods works.