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# 16 - 16 Numerical Integration 16 Formulae Chapter...

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16: Numerical Integration 16: Numerical Integration Formulae Formulae

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Nov 15, 2006 23:10 ECOR2606 -- Hassan & Holtz 2 Chapter Objectives Introduction to numerical integration : Newton-Cotes formulae – replace complex function with interpolating polynomial, and integrate the polynomial. implement the following single-application Newton-Cotes formulae: Trapezoidal rule Simpson's 1/3 rule Simpson's 3/8 rule implement the composite Newton-Cotes formulae: Trapezoidal rule Simpson's 1/3 rule Simpson's 3/8 rule (NIT) know that even-segment-odd-point formulae (like Simpson's 1/3) achieve higher than expected accuracy. know how to use trapezoidal rule for unequally spaced data. know the difference between open and closed integration formulae.
Nov 15, 2006 23:10 ECOR2606 -- Hassan & Holtz 3 16.1: Introduction and Background The definite integral: Some typical applications: Figure 16.1 I = a b f x dx area, I Figure 16.2

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Nov 15, 2006 23:10 ECOR2606 -- Hassan & Holtz 4 Introduction and Background (2) Mean values: discrete: continuous: Centroids, centres of gravity: Mean = i = 1 n y i n Mean = a b f x dx b a Figure 16.3 x = a b x f x dx a b f x dx
Nov 15, 2006 23:10 ECOR2606 -- Hassan & Holtz 5 16.2: Newton-Cotes Formulae The most common numerical integration scheme. To determine where this is difficult or impossible to do analytically: evaluate the function at n+1 points to get the discrete data: determine an order n interpolating polynomial integrate the polynomial: to reduce errors, limit the order, n , and the range, b-a . subdivide range a-b into subranges: x 0 -x 1 , x 1 -x 2 , ..., x m-1 -x m I = a b f x dx x 1, f x 1 , x 2, f x 2 , x n 1 , f x n 1 I a b f n x dx f n x  = a 0 a x  ⋯  a n 1 x n 1 a n x n (16.9) error in text

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Nov 15, 2006 23:10 ECOR2606 -- Hassan & Holtz 6 Newton-Cotes Formulae (2) (a) 1 st order polynomial (b) 2 nd order polynomial 3 1 st order polynomials piecewise – sum the areas of each piece. Figure 16.4 Figure 16.5
Nov 15, 2006 23:10 ECOR2606 -- Hassan & Holtz 7 Newton-Cotes Formulae (3) (a) closed form : data points at beginning and end of limits of integration are known. closed form methods are emphasized here. Figure 16.6 (b) open form : limits of integration extend beyond range of data points.

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Nov 15, 2006 23:10 ECOR2606 -- Hassan & Holtz 8 16.3: Trapezoidal Rule 1 st order case : the area of the trapezoid all Newton-Cotes rules are similar “average height” computation differs f 1 x = f a   f b − f a b a x a I = a b f 1 x dx I =  b a f a  f b 2 (16.11) Figure 16.7 I =  b a × average height
Nov 15, 2006 23:10 ECOR2606 -- Hassan & Holtz 9 Trapezoidal Rule (2) 16.3.1: Error of the Trapezoidal Rule : Estimate of truncation error for a single application: to attempt to determine a numeric value for this, use either: a mean value, or minimum and maximum values of if that can be determined.

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