Chapter 5: Numerical Integration and Differentiation

Chapter 5: Numerical Integration and DifferentiationPART I: Numerical IntegrationNewton-Cotes Integration FormulasThe idea of Newton-Cotes formulas is to replace a complicated function or tabu-lated data with an approximating function that is easy to baf(x)dx bafn(x)dxwherefn(x) =a0+a1x+a2x2+..+ trapezoidal RuleUsing the first order Taylor series to approximatef(x),I= baf(x)dx baf1(x)dxwheref1(x) =f(a) +f(b) f(a)b a(x a)1 ThenI ba[f(a) +f(b) 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 trapezoidalFigure 1: Graphical depiction of the trapezoidal ruleunder the straight line connectingf(a)andf(b). An estimate for the local trun-2cation error of a single application of the trapezoidal rule can be obtained usingTaylor series asEt= 112f ( )(b a)3where is a value :Use the trapezoidal rule to numerically integratef(x) = + 25xfroma= 0tob= :f(a) =f(0) = , andf(b) =f(2) = (b a)f(b) +f(a)2= (2 0) + true solution is 20f(x)dx= ( + )|20= ( 2 + 22) 0 = (x)is a linear function, using the trapezoidal rule gets the exact :Use the trapezoidal rule to numerically integratef(x) = + 25x+ 3x23froma= 0tob= :f(0) = , andf(2) = (b a)f(b) +f(a)2= (2 0) + true solution is 20f(x)dx= ( + +x3)|20= ( 2 + 22+ 23) 0 = relative error is| t|= 100% = trapezoidal rule:Using smaller Integration interval can reduce the approximation error.

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

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  Rules, Trapezoidal, Trapezoidal rule

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