mws_gen_int_ppt_simpson13(1)

mws_gen_int_ppt_simpson13(1) - 1/10/2010...

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Unformatted text preview: 1/10/2010 http://numericalmethods.eng.usf.edu 1 Simpson’s 1/3 rd Rule of Integration Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates Simpson’s 1/3 rd Rule of Integration http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu 3 What is Integration? Integration ∫ = b a dx ) x ( f I The process of measuring the area under a curve. Where: f(x) is the integrand a= lower limit of integration b= upper limit of integration f(x) a b y x ∫ b a dx ) x ( f http://numericalmethods.eng.usf.edu 4 Simpson’s 1/3 rd Rule http://numericalmethods.eng.usf.edu 5 Basis of Simpson’s 1/3 rd Rule Trapezoidal rule was based on approximating the integrand by a first order polynomial, and then integrating the polynomial in the interval of integration. Simpson’s 1/3rd rule is an extension of Trapezoidal rule where the integrand is approximated by a second order polynomial. Hence ∫ ∫ ≈ = b a b a dx ) x ( f dx ) x ( f I 2 Where is a second order polynomial. ) x ( f 2 2 2 1 2 x a x a a ) x ( f + + = http://numericalmethods.eng.usf.edu 6 Basis of Simpson’s 1/3 rd Rule Choose )), a ( f , a ( , b a f , b a             + + 2 2 )) b ( f , b ( and as the three points of the function to evaluate a , a 1 and a 2 . 2 2 1 2 a a a a a ) a ( f ) a ( f + + = = 2 2 1 2 2 2 2 2       + +       + + =       + =       + b a a b a a a b a f b a f 2 2 1 2 b a b a a ) b ( f ) b ( f + + = = http://numericalmethods.eng.usf.edu 7 Basis of Simpson’s 1/3 rd Rule Solving the previous equations for a , a 1 and a 2 give 2 2 2 2 2 2 4 b ab a ) a ( f b ) a ( abf b a abf ) b ( abf ) b ( f a a + − + +       + − + = 2 2 1 2 2 4 3 3 2 4 b ab a ) b ( bf b a bf ) a ( bf ) b ( af b a af ) a ( af a + − +       + − + +       + − − = 2 2 2 2 2 2 2 b ab a ) b ( f b a f ) a ( f a + −       +       + − = http://numericalmethods.eng.usf.edu 8 Basis of Simpson’s 1/3 rd Rule Then ∫ ≈ b a dx ) x ( f I 2 ( ) ∫ + + = b a dx x a x a a 2 2 1 b a x a x a x a       + + = 3 2 3 2 2 1 3 2 3 3 2 2 2 1 a b a a b a ) a b ( a − + − + − = http://numericalmethods.eng.usf.eduhttp://numericalmethods....
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida - Tampa.

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mws_gen_int_ppt_simpson13(1) - 1/10/2010...

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