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Chapter 20. Integral Calculus - Pg. 633

Integral Calculus Introduction 633 20.1 Integration 634 20.1.1 Indefinite Integration 634 Integrate 20.1.2 More about Indefinite Integration 637 20.1.3 Definite Integration 638 Integrate 20.1.4 Integration over Regions 642 Boole 20.2 Numerical Quadrature 644 20.2.1 Introduction 644 NIntegrate 20.2.2 Strategies and Rules 647 20.2.3 General Adaptive Strategies 648 "GlobalAdaptive", "LocalAdaptive" 20.2.4 Quadrature Rules 652 "GaussKronrodRule" , "MultiDimensionalRule", etc. 20.2.5 Singular Integrands 655 IMT , "DoubleExponential" , "DuffyCoordinates" 20.2.6 Oscillatory Integrands 658 "ClenshawCurtisRule" , "ExtrapolatingOscillatory", etc. 20.2.7 Symbolic Preprocessing 660 "SymbolicPiecewiseSubdivision" , "EvenOddSubdivision", etc. 20.2.8 Monte Carlo Methods 662 "MonteCarlo" , "AdaptiveMonteCarlo", etc. 20.2.9 More about Quadrature 664 NewtonCotesWeights, GaussianQuadratureWeights, etc. 20.3 Sums and Products 666 20.3.1 Exact Sums 666 Sum 20.3.2 Numerical Sums 667 NSum 20.3.3 Products 670 Product, NProduct 20.4 Transforms 670 20.4.1 Laplace Transforms 670 LaplaceTransform, InverseLaplaceTransform 20.4.2 Z-Transforms 672 ZTransform, InverseZTransform 20.4.3 Fourier Transforms and Series 672 FourierTransform, FourierTrigSeries, etc. 20.4.4 Discrete Fourier Transforms 675 Fourier, InverseFourier, FourierDST, FourierDCT Introduction Riemann consulted a doctor about his diet. He was told to reduce the amount of food he ate at each meal but to increase the number of meals. He proceeded to do so and ultimately ate infinitesimal amount infinitely often, and he found that his weight did not change. Shortly after this, he gave a precise definition of a definite integral. Mathematica's symbolic integration methods have such power and knowledge that in the Documenta- tion Center it is said that Integrate can evaluate essentially all indefinite integrals and most definite integrals listed in standard books of tables.