Chapter 15Numerical IntegrationThe need to integrate a functionf(x) arises often in mathematical modeling. This chapter is devotedto techniques for approximating this operation numerically.Such techniques are at the heart ofapproximation methods for solving differential and integral equations, which are important andactive areas of research.Note:The material covered here is classical, but we do not settle for just presenting a se-quence of integration rules. Rather, we take the opportunity to introduce several importantconcepts and approaches that extend well beyond numerical integration without having togo into complicating or confusing detail. This is the main reason for the present chapter’sconsiderable length.We concentrate on definite integrals. Thus, unless otherwise noted we seek to approximateIf=baf(x)dx≈nj=0ajf(xj)(15.1)for a givenfinite interval [a,b] and an integrable functionf. The numerical integration formula,often referred to as aquadrature rule, hasabscissae xjandweights aj.In Section 15.1 we start off by deriving basic rules for numerical integration based on, guesswhat, polynomial interpolation. Then the arguments that led in previous chapters from polynomialinterpolation to piecewise polynomial interpolation apply again: these basic rules are good locallybut often not globally when the integration interval [a,b] is not necessarily short. The compositeformulas developed in Section 15.2 are similar in spirit to those of Chapter 11, although they aresimpler here.Sections 15.3–15.5 introduce more advanced concepts that arise, in a far more complex form,in numerical methods for differential equations. When designing basic rules for numerical inte-gration, it is generally possible to achieve a higher accuracy order than is possible for numericaldifferentiation based on the same points. This is exploited in Section 15.3, where clever choices ofabscissae are considered, resulting inGaussian quadrature. Another approach calledRombergintegrationfor obtaining high order formulas, extending the extrapolation methods of Section 14.2,is developed in Section 15.5.In Section 15.4 an essential premise changes. We no longer seek just formulas that behavewell, in principle, when some parameterhis“small enough.”Rather, a general-purpose program is441Downloaded 12/08/18 to 132.174.255.3. Redistribution subject to SIAM license or copyright; see

442Chapter 15. Numerical Integrationdesigned that delivers approximations for definite integrals that are within a user-specified toleranceof the exact value.Finally, numerical integration in more than one dimension is briefly considered in the advancedSection 15.6. When there are many dimensions, the corresponding numerical methods change radi-cally.