Mathematische Formelsammlung

11.8 Numerische Integration a = x 0 < x 1 < x 2 < … < x n – 1 < x n = b; x i – x i – 1 = ​  b – a __ n  ​= ∆ x für i = 1, 2, 3, …, n Rechtecksformel ​ :  a ​  b ​ f(x)​dx ≈ ​  b – a _ n ​ ​ [ f(x 0 ) + f(x 1 ) + f(x 2 ) + … + f(x n – 1 ) ] ​= = ∆ x · ​ ;  i = 0 ​  n – 1 ​ f(x i )​ Trapezformel ​ :  a ​  b ​ f(x)​dx ≈ ​  b – a _ 2n ​ ​ [ f(x 0 ) + 2 f(x 1 ) + 2 f(x 2 ) + … +  ​ ​ ​ ​  + 2 f(x n – 1 ) + f(x n ) ] ​ Regel von Simpson a = x 0 < x 1 < x 2 < … < x 2n – 1 < x 2n = b; x i – x i – 1 = ​  b – a __ 2n ​für i = 1, 2, 3, …, 2n ​ :  a ​  b ​ f(x)​dx ≈ ​  b – a _ 6n ​​ {  f(x 0 ) + f(x 2n ) + 2 ​ [ f(x 2 ) + f(x 4 ) + f(x 6 ) +   ​ ​  ​ ​ ​ ​  + … + f(x 2n – 2 ) ] ​+ 4 ​ ​  ​ [ f(x 1 ) + f(x 3 ) + f(x 5 ) + … + f(x 2n – 1 ) ] ​ } ​ Regel von Kepler (Fassregel) ​ :  a ​  b ​ f(x)​dx ≈ ​  b – a _ 6 ​​ [ f(a) + 4f ​ (  ​  a + b _ 2 ​  ) ​+ f(b) ] ​ x 0 f b a x y ∆ x 0 f b a y x 0 f b a y x 0 f b a a+b 2 y Differential- und Integralrechnung 29 Nur zu Prüfzwecken – Eigentum des Verlags öbv