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www.le.ac.uk Numerical Methods: Integration Department of Mathematics University of Leicester

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Content MotivationMid-ordinate ruleSimpson’s rule

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Reasons for Numerical Integration The function could be difficult or impossible to integrate The function may have been obtained from data, and the function may not be known We can program a computer to approximate any integral using numerical integration Next Mid-ordinate rule Simpson’s rule Motivation

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Numerical Integration: Mid-ordinate rule The Mid-ordinate rule is a numerical way of finding the area under a curve, by dividing the area into rectangles Next Mid-ordinate rule Simpson’s rule Motivation

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Numerical Integration: Mid-ordinate rule h AB Mid-ordinate rule Simpson’s rule Motivation Next

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Numerical Integration: Mid-ordinate rule Mid-ordinate rule Simpson’s rule Motivation Next

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So the area under the curve is approximately: Numerical Integration: Mid-ordinate rule Mid-ordinate rule Simpson’s rule Motivation Next

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Numerical Integration: Mid-ordinate rule Mid-ordinate rule Simpson’s rule Motivation Next

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Firstly, we calculate the width of the strips Numerical Integration: Mid-ordinate rule Mid-ordinate rule Simpson’s rule Motivation Next

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Numerical Integration: Mid-ordinate rule xyEvaluate 0.05y1y1 1.05127 0.15y2y2 1.16834 0.25y3y3 1.28403 0.35y4y4 1.41907 0.45y5y5 1.56831 0.55y6y6 1.73325 0.65y7y7 1.91554 0.75y8y8 2.11700 0.85y9y9 2.33965 0.95y 10 2.58571 TOTAL17.18217 Mid-ordinate rule Simpson’s rule Motivation Next

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Using the formula we get Numerical Integration: Mid-ordinate rule Next Mid-ordinate rule Simpson’s rule Motivation

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4 2 0 5 31 Mid-Ordinate Rule Next Mid-ordinate rule Simpson’s rule Motivation

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Numerical Integration: Simpson’s rule Simpson’s rule is a form of numerical integration which uses quadratic polynomials.... Next Mid-ordinate rule Simpson’s rule Motivation

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We then approximate the areas of the pairs of strips in the following way Numerical Integration: Simpson’s rule Next Mid-ordinate rule Simpson’s rule Motivation

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Numerical Integration: Simpson’s rule Next Mid-ordinate rule Simpson’s rule Motivation

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Numerical Integration: Simpson’s rule xyFirst and LastOddEven 01 0.11.010 0.21.040 0.31.094 0.41.173 0.51.284 0.61.433 0.71.632 0.81.896 0.92.247 12.718 TOTAL3.7187.2685.544 Next Mid-ordinate rule Simpson’s rule Motivation

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Numerical Integration: Simpson’s rule Next Mid-ordinate rule Simpson’s rule Motivation

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4 2 0 5 31 Simpson’s Rule Next Mid-ordinate rule Simpson’s rule Motivation

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