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.

11.1Sample 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
11.2Evaluating 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
11.3Evaluating 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
11.4Evaluating 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
11.5Algorithms 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
11.6Evaluating 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
11.7Evaluating 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
11.8Evaluating 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
11.9Evaluating 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
11.10Evaluating 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
11.11Evaluating Integrals Numerically with FORTRAN
11.11.1 Writing Programs from Scratch
11.11.2 Using Numerical Recipes
11.12Evaluating Integrals Numerically with C
11.12.1 Writing Programs from Scratch
11.12.2 Using Numerical Recipes
11.13Exercises
11.13.1 ... using Symbolic Methods
11.13.2 ... using Numerical Methods
11.13.3 ... using Numerical Recipes
11.AListing of trapezoidal.f
11.BListing of trapezoidal.c

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