Transcription of Numerical Analysis: Trapezoidal and Simpson's Rule
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Numerical Analysis: Trapezoidal and Simpson's Rule
NumericalAnalysis:Trapezoidaland Simpson sRuleNatasha , PhDNumerical analysis : Trapezoidal andSimpson s RuleNatasha S. Sharma, PhDNumericalAnalysis:Trapezoidaland Simpson sRuleNatasha , PhDMathematical question we are interested innumerically answeringHow to we evaluateI= baf(x)dx?Calculus tells us that if F(x) is the antiderivative of afunctionf(x) on the interval [a,b], thenI= baf(x)dx=F(x)|ba=F(b) F(a).Practically, most integrals cannot be evaluated using thisapproach. For example, 10dx1 +x5has a complicated antiderivative and it easier to adopt anumerical method to approximate this :Trapezoidaland Simpson sRuleNatasha , PhDNumerical Integration: A General FrameworkIf you cannot solve a problem, then replace it with a near-by problem that you can solve!
GoalChoose an approximation ~f(x) to f(x) that is easily integrable and a good approximation to f(x). Two natural candidates: 1 Taylor polynomials approximating f(x). One caveat: We need f(x) to have derivatives at \a" to exist of a higher order to improve the approximation! 2 Interpolating polynomials approximating f(x).
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