Chapter5: Numerical Integration and Differentiation
         PARTI:NumericalIntegration
         Newton-Cotes Integration Formulas
         TheideaofNewton-Cotesformulasistoreplaceacomplicatedfunctionortabu-
         lated data with an approximating function that is easy to integrate.
                                I = Z bf(x)dx ≈ Z bf (x)dx
                                                      n
                                      a            a
                                      2           n
         where f (x) = a +a x+a x +...+a x .
                 n       0   1     2            n
         1  TheTrapezoidalRule
         Using the first order Taylor series to approximate f(x),
                                I = Z bf(x)dx ≈ Z bf (x)dx
                                                      1
                                      a            a
         where
                              f (x) = f(a) + f(b) − f(a)(x − a)
                               1                b −a
                                              1
         Then                  Z ·                         ¸
                                 b        f(b) −f(a)
                          I ≈ a f(a)+        b −a   (x−a) dx
                            = (b−a)f(b)+f(a)
                                          2
          The trapezoidal rule is equivalent to approximating the area of the trapezoidal
                               Figure 1: Graphical depiction of the trapezoidal rule
         under the straight line connecting f(a) and f(b). An estimate for the local trun-
                                            2
          cation error of a single application of the trapezoidal rule can be obtained using
          Taylor series as
                                    E =−1f′′(ξ)(b−a)3
                                     t     12
          where ξ is a value between a and b.
          Example: Use the trapezoidal rule to numerically integrate
                                       f(x) = 0.2 + 25x
          from a = 0 to b = 2.
          Solution: f(a) = f(0) = 0.2, and f(b) = f(2) = 50.2.
                    I = (b −a)f(b)+f(a) = (2−0)× 0.2+50.2 = 50.4
                                    2                      2
          Thetrue solution is
                Z 2f(x)dx = (0.2x+12.5x2)|2 = (0.2×2+12.5×22)−0 = 50.4
                                            0
                 0
          Because f(x) is a linear function, using the trapezoidal rule gets the exact solu-
          tion.
          Example: Use the trapezoidal rule to numerically integrate
                                                         2
                                    f(x) = 0.2 + 25x + 3x
                                              3
         from a = 0 to b = 2.
         Solution: f(0) = 0.2, and f(2) = 62.2.
                    I = (b − a)f(b) + f(a) = (2 − 0) × 0.2 + 62.2 = 62.4
                                   2                     2
         Thetrue solution is
          Z 2                      2   3 2                    2   3
           0 f(x)dx = (0.2x+12.5x +x )|0 = (0.2×2+12.5×2 +2 )−0 = 58.4
         Therelative error is    ¯          ¯
                                 ¯58.4 − 62.4¯
                            |ǫt| = ¯        ¯ ×100% = 6.85%
                                 ¯   58.4   ¯
                                            4