# Chapter 11 Evaluating Integrals

Note: While inclusion of all components in this chapter would, in principle, be possible, selection of only a few of MACSYMA, MAPLE, Mathematica, IDL, MATLAB, FORTRAN (with or without Numerical Recipes), and C (with or without Numerical Recipes) would probably be more common.

 11.1 Sample Problems 11.1.1 One-Dimensional Trajectories 11.1.2 Center of Mass 11.1.3 Moment of Inertia 11.1.4 The Large Amplitude Simple Pendulum 11.1.5 Statistical Data Analysis 11.1.6 The Cornu Spiral 11.1.7 Electric and Magnetic Fields and Potentials 11.1.8 Quantum Probabilities 11.1.9 Expansion in Orthogonal Functions 11.2 Evaluating Integrals Symbolically with Macsyma 11.2.1 Relativistic Motion Under a Constant Force 11.2.2 Center of Mass 11.2.3 Moment of Inertia; Radius of Gyration 11.2.4 Electrostatic Potential of a Finite Line Charge 11.2.5 The Helmholtz Coil 11.2.6 Period of a Pendulum with Modest Amplitude 11.2.7 Fourier Coefficients for Half-Rectified Signal 11.3 Evaluating Integrals Symbolically with Maple 11.3.1 Relativistic Motion Under a Constant Force 11.3.2 Center of Mass 11.3.3 Moment of Inertia; Radius of Gyration 11.3.4 Electrostatic Potential of a Finite Line Charge 11.3.5 The Helmholtz Coil 11.3.6 Period of a Pendulum with Modest Amplitude 11.3.7 Fourier Coefficients for Half-Rectified Signal 11.4 Evaluating Integrals Symbolically with Mathematica 11.4.1 Relativistic Motion Under a Constant Force 11.4.2 Center of Mass 11.4.3 Moment of Inertia; Radius of Gyration 11.4.4 Electrostatic Potential of a Finite Line Charge 11.4.5 The Helmholtz Coil 11.4.6 Period of a Pendulum with Modest Amplitude 11.4.7 Fourier Coefficients for Half-Rectified Signal 11.5 Algorithms for Numerical Integration 11.5.1 Newton-Cotes Quadrature 11.5.2 Rearrangements for Computational Efficiency 11.5.3 Assessing Error 11.5.4 Iterative and Adaptive Algorithms 11.5.5 Gaussian Quadrature 11.6 Evaluating Integrals Numerically with IDL 11.6.1 Using Elementary Commands 11.6.2 The Function luqsimp 11.6.3 Moment of Inertia 11.6.4 Quantum Probabilities 11.6.5 Integrals as Functions of the Upper Limit 11.6.6 The Error Function 11.6.7 The Cornu Spiral 11.6.8 Integrals as Functions of an Internal Parameter 11.6.9 The Off-Axis Electrostatic Potential of Two Rings 11.7 Evaluating Integrals Numerically with MATLAB 11.7.1 Using Elementary Commands 11.7.2 The Functions trapz, quad, quad8and quadl 11.7.3 Moment of Inertia 11.7.4 Quantum Probabilities 11.7.5 Integrals as Functions of the Upper Limit 11.7.6 The Error Function 11.7.7 The Cornu Spiral 11.7.8 Integrals as Functions of an Internal Parameter 11.7.9 The Off-Axis Electrostatic Potential of Two Rings 11.8 Evaluating Integrals Numerically with MACSYMA 11.8.1 Romberg and Bromberg Integration 11.8.2 Newton-Cotes Quadrature 11.8.3 Quantum Probability 11.8.4 The Error Function 11.8.5 The Off-Axis Electrostatic Potential of Two Rings 11.9 Evaluating Integrals Numerically with MAPLE 11.9.1 Quantum Probability 11.9.2 The Error Function 11.9.3 The Off-Axis Electrostatic Potential of Two Rings 11.10 Evaluating Integrals Numerically with Mathematica 11.10.1 Quantum Probability 11.10.2 The Error Function 11.10.3 The Off-Axis Electrostatic Potential of Two Rings 11.11 Evaluating Integrals Numerically with FORTRAN 11.11.1 Writing Programs from Scratch 11.11.2 Using Numerical Recipes 11.12 Evaluating Integrals Numerically with C 11.12.1 Writing Programs from Scratch 11.12.2 Using Numerical Recipes 11.13 Exercises 11.13.1 ... using Symbolic Methods 11.13.2 ... using Numerical Methods 11.13.3 ... using Numerical Recipes 11.A Listing of trapezoidal.f 11.B Listing of trapezoidal.c