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1/10/2010 http://numericalmethods.eng.usf.edu 1 Gauss Quadrature Rule of Integration Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

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Gauss Quadrature 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

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http://numericalmethods.eng.usf.edu 4 Two-Point Gaussian Quadrature Rule
http://numericalmethods.eng.usf.edu 5 Basis of the Gaussian Quadrature Rule Previously, the Trapezoidal Rule was developed by the method of undetermined coefficients. The result of that development is summarized below. ) ( 2 ) ( 2 ) ( ) ( ) ( 2 1 b f a b a f a b b f c a f c dx x f b a + = +

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http://numericalmethods.eng.usf.edu 6 Basis of the Gaussian Quadrature Rule The two-point Gauss Quadrature Rule is an extension of the Trapezoidal Rule approximation where the arguments of the function are not predetermined as a and b but as unknowns x 1 and x 2 . In the two-point Gauss Quadrature Rule, the integral is approximated as = b a dx ) x ( f I ) x ( f c ) x ( f c 2 2 1 1 +
http://numericalmethods.eng.usf.edu 7 Basis of the Gaussian Quadrature Rule The four unknowns x 1 , x 2 , c 1 and c 2 are found by assuming that the formula gives exact results for integrating a general third order polynomial, . x a x a x a a ) x ( f 3 3 2 2 1 0 + + + = Hence ( ) + + + = b a b a dx x a x a x a a dx ) x ( f 3 3 2 2 1 0 b a x a x a x a x a + + + = 4 3 2 4 3 3 2 2 1 0 ( ) + + + = 4 3 2 4 4 3 3 3 2 2 2 1 0 a b a a b a a b a a b a

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http://numericalmethods.eng.usf.edu 8 Basis of the Gaussian Quadrature Rule It follows that ( ) ( ) 3 2 3 2 2 2 2 1 0 2 3 1 3 2 1 2 1 1 0 1 x a x a x a a c x a x a x a a c dx ) x ( f b a + + + + + + + = Equating Equations the two previous two expressions yield ( ) + + + 4
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