10 NumericalIntegration

10 NumericalIntegration - Numerical Integration and...

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Numerical Integration and Differentiation CEE3804: Computer Apps for Civil and Environmental Engineering
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Newton Cotes Integration Formulas z Replace function or tabulated data to be integrated with a simpler, easy to integrate approximating function z where f n (x) is a polynomial of form ∫∫ = b a b a n dx x f dx x f I ) ( ) ( n n n x a x a x a a x f + + + + = ... ) ( 2 2 1 0
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Trapezoid Rule z Newton-Cotes with n = 1 z where ∫∫ = b a b a dx x f dx x f I ) ( ) ( 1 () a x a b a f b f a f x f + = ) ( ) ( ) ( ) ( 1
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Chapra
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Trapezoid Rule, continued z Substituting and integrating z This is termed the trapezoid rule z For all Newton-Cotes I = width * average height where the average height varies with n () 2 ) ( ) ( b f a f a b I + =
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Chapra
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Trapezoid Rule, Comments z Easy to implement z Works with function or tabulated data z For equally spaced data with n equally spaced points, formula can be simplified to z But relatively poor accuracy compared to other techniques. Accuracy increases as use more points (i.e. delta x smaller) x y y y I n i i n + + = = 1 2 1 2
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Trapezoid Rule Examples in Excel z Trapeziod unequal.xls z Trapezoid equal.xls
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10 NumericalIntegration - Numerical Integration and...

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