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

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Presentation on theme: "Www.le.ac.uk Numerical Methods: Integration Department of Mathematics University of Leicester."— Presentation transcript:

1 Numerical Methods: Integration Department of Mathematics University of Leicester

2 Content MotivationMid-ordinate ruleSimpson’s rule

3 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

4 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

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

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

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

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

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

10 Numerical Integration: Mid-ordinate rule xyEvaluate 0.05y1y y2y y3y y4y y5y y6y y7y y8y y9y y TOTAL Mid-ordinate rule Simpson’s rule Motivation Next

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

12 Mid-Ordinate Rule Next Mid-ordinate rule Simpson’s rule Motivation

13 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

14 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

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

16 Numerical Integration: Simpson’s rule xyFirst and LastOddEven TOTAL Next Mid-ordinate rule Simpson’s rule Motivation

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

18 Simpson’s Rule Next Mid-ordinate rule Simpson’s rule Motivation

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