Numerical Differentiation and Integration

This chapter gives introduction to numerical differentiation by means of an expansion into a Taylor series and interpolation polynomials, and numerical integration. The numerical integration formulas include the Newton-CôTes quadrature formulae, the trapezoid formula, Simpson's formula, Euler's and Gregory's formulae, Romberg's formula, and Chebyshev's quadrature formulae. In addition, the chapter considers quadrature formulae of Gauss type obtained by orthogonal polynomials, calculation of improper integrals, Kantorovich's method, and the Monte Carlo method for calculation of definite integrals. These are followed by applications.