Numerical integration, which is also called quadrature, has a history extending back to the invention of calculus and before. The fact that integrals of elementary functions could not, in general, be computed analytically, while derivatives could be, served to give the field a certain panache, and to set it a cut | Chapter 4. Integration of Functions Introduction Numerical integration which is also called quadrature has a history extending back to the invention of calculus and before. The fact that integrals of elementary functions could not in general be computed analytically while derivatives could be served to give the field a certain panache and to set it a cut above the arithmetic drudgery of numerical analysis during the whole of the 18th and 19th centuries. With the invention of automatic computing quadrature became just one numerical task among many and not a very interesting one at that. Automatic computing even the most primitive sort involving desk calculators and rooms full of computers that were until the 1950s people rather than machines opened to feasibility the much richer field of numerical integration of differential equations. Quadrature is merely the simplest special case The evaluation of the integral I f x dx J a is precisely equivalent to solving for the value I y b the differential equation dy f x dx with the boundary condition y a 0 Chapter 16 of this book deals with the numerical integration of differential equations. In that chapter much emphasis is given to the concept of variable or adaptive choices of stepsize. We will not therefore develop that material here. If the function that you propose to integrate is sharply concentrated in one or more peaks or if its shape is not readily characterized by a single length-scale then it is likely that you should cast the problem in the form of - and use the methods of Chapter 16. The quadrature methods in this chapter are based in one way or another on the obvious device of adding up the value of the integrand at a sequence of abscissas within the range of integration. The game is to obtain the integral as accurately as possible with the smallest number of function evaluations of the integrand. Just as in the case of interpolation Chapter 3 one has the freedom to choose .
