Integration

This chapter covers different aspects of integration, focusing on numerical integration. For historical reasons, numerical integration is also known as quadrature. Integration is significantly more complex than its inverse operation—differentiation—and while many examples of integrals can be calculated analytically, in general, we must resort to numerical methods. Depending on the properties of the integrand (the function being integrated) and the integration limits, it can be easy or difficult to compute an integral numerically. In most cases, integrals of continuous functions and finite integration limits can be computed efficiently in one dimension. But integrable functions with singularities or integrals with infinite integration limits are cases that can be difficult to handle numerically, even in a single dimension. Two-dimensional integrals (double integrals) and higher-order integrals can be numerically computed with repeated single-dimension integration or using multidimensional generalizations of the techniques used to solve single-dimensional integrals. However, the computational complexity grows quickly with the number of dimensions to integrate. Such methods are only feasible for low-dimensional integrals like double or triple integrals. Integrals of higher dimensions than that often require entirely different techniques, such as Monte Carlo sampling algorithms.