mws_gen_int_ppt_simpson13(1)

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

This preview shows pages 1–10. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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.

### Page1 / 32

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

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online