2018 | OriginalPaper | Chapter

# Numerical Integration

Authors: Michael Oberguggenberger, Alexander Ostermann

Publisher: Springer International Publishing

Log inThe fundamentalIntegrationnumerical theorem of calculus suggests the following approach to the calculation of definite integrals: One determines an antiderivative F of the integrand f and computes from that the value of the integral $$ \int ^b_a \! f(x) \, \mathrm{d}x = F(b) - F(a). $$ ∫ a b f ( x ) d x = F ( b ) - F ( a ) . In practice, however, it is difficult and often even impossible to find an antiderivative F as a combination of elementary functions. Apart from that, antiderivatives can also be fairly complex, as the example $$\int x^{100} \sin x \,\mathrm{d}x$$ ∫ x 100 sin x d x shows. Finally, in concrete applications the integrand is often given numerically and not by an explicit formula. In all these cases one reverts to numerical methods. In this chapter the basic concepts of numerical integration (quadrature formulas and their order) are introduced and explained. By means of instructive examples we analyse the achievable accuracy for the Gaussian quadrature formulas and the required computational effort.