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Chapter 7 Numerical Integration 101109

# Chapter 7 Numerical Integration 101109 - Chapter 7...

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1 Chapter 7 Numerical Integration Contents Introduction ......................................................................................................................... 3 7.1 The Trapezoidal Rule .................................................................................................... 3 7.1.1 A Geometric Interpretation of the Trapezoidal Rule ............................................. 4 7.1.2 An Expression for the Trapezoidal Rule Error Term ............................................. 6 Example 7.1 Numerical Integration with the Trapezoidal Rule ..................................... 9 7.2 Trapezoidal Rule with End Correction ....................................................................... 15 Example 7.2 Trapezoidal Rule with End Correction .................................................... 16 7.3 Simpson’s 1/3 Rule ..................................................................................................... 18 7.3.1 Derivation of Simpson's 1/3 Rule ........................................................................ 18 Example 7.3 Applying Simpson’s 1/3 Rule .................................................................. 20 7.3.2 User Defined Functions Simpson(a,b,n) and Simpson_sf(a,b,sf) ........................ 21 Example 7.4 Comparing Trapezoidal Rule, Trapezoidal Rule with End Correction, and Simpson's 1/3 Rule .................................................................................. 22 7.4 Richardson Extrapolation and Romberg Integration .................................................. 23 Example 7.4 Romberg Integration for the Error Function in a User Defined Function with VB ................................................................................................... 26 7.5 Gauss-Legendre Quadrature over Subintervals .......................................................... 29 7.5.1 Derivation of the Gauss-Legendre Quadrature Expression ................................. 29 Example 7.5 Integration of I=0 1x4dx using Gauss-Legendre Quadrature .................. 32 Example 7.6 Error Function Analysis with Excel and Gauss-Legendre Quadrature ... 33 7.5.2 Implementing Gauss-Legendre Quadrature in a User Defined Function ............ 35

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2 7.6 More Gaussian Quadratures, Improper Integrals, and Singularities ........................... 37 7.7 Chapter Summary ....................................................................................................... 37 Exercises ........................................................................................................................... 38 Table of Figures Figure 7.1 Approximating the Area under a Curve with a Series of Trapezoids ................ 4 Figure 7.2 Approximating the Integral with a Single Panel and the Trapezoid Rule ......... 9 Figure 7.3 Improving the Approximation of the Definite Integral by using Two Panels . 10 Figure 7.4 Improving the Trapezoidal Rule Approximation with Three Panels .............. 11 Figure 7.5 Spreadsheet Implementation of the Trapezoidal Rule for Example 7.1 .......... 11 Figure 7.6 VB Function to Implement the Trapezoidal Rule as a User Function ............ 12 Figure 7.7 Converging Numerical Integration in Example 7.1 ........................................ 13 Figure 7.8 Cell Formulas used for Spreadsheet in Figure 7.7 .......................................... 13 Figure 7.9 VB Code for Function to Converge an Integration with the Trapezoidal Rule14 Figure 7.10 Finding the Number of Panels to Converge Example 7.1 to the Specified Significant Figures with the Trapezoidal Rule ................................................................. 15 Figure 7.11 Modifying the Trapezoidal Rule Programs to Add End Correction .............. 16 Figure 7.12 Comparing Convergence Performance of Trapezoidal Rule vs Trapezoidal Rule with End Correction ................................................................................................. 17 Figure 7.13 Approximating the Integral of ࢌሺ࢞ሻ with the Integral of the Lagrange Polynomial ሺ࢞ሻ ............................................................................................................. 19 Figure 7.14 Simpson's 1/3 rd Rule for Example 7.3 with 2 Panels .................................... 20 Figure 7.15 VB Code for a User Defined Function for Simpson's 1/3 Rule .................... 22 Figure 7.16 Comparison of Convergence Behavior for Trapezoidal Rule, Trapezoidal Rule with End Correction, and Simpson's 1/3rd Rule ...................................................... 23 Figure 7.17 Using Richardson Extrapolation with the Trapezoidal Rule ......................... 24 Figure 7.18 Elements of a Romberg Integration Table ..................................................... 25 Figure 7.19 Romberg Integration Table for ࡵ ൌ ׬ ݔ ࢊ࢞ ................................................ 26 Figure 7.20 VB Code for User Defined Function Romberg( a,b,sf ) . .............................. 27 Figure 7.21 Modules used for Example 7.4 Error Function Evaluation ........................... 28 Figure 7.22 Error Function Calculations and Romberg Table Generation using Romberg( a,b,sf ) .............................................................................................................. 28 Figure 7.23 Numerical Evaluation of the Error Function Integral
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