Theorem 2.1(Closed Newton-Cotes Quadrature Formula). Assume that k =co+ kh are equally spaced nodes and fr= f(k). The first fo our closed Newton-Cotes quadrature formulas are 1 f(x)da≈x(fo+f1) (the trapezoidal rule) f(a)d. c s(fo +4f1+f2)(Si lmpson s rule (2.5) 厂f 3h (a)d ac a o(fo+3f 1+3f2+ f3)(Simpsons grule),(2.6) 2h f(x)dr≈:(7+32f1+12f2+32g+7f4)( Boole rule.(2.7) Figure 2.2(a) The trapezoidal rule integrates(b) Sinpson's rule integrates(c) Simpsons rule integrates(d) Boole's rule integrates