Although one has the impression that the integration is the counterpart of differentiation, integration is much more difficult. To determine derivatives of specific functions, it is sufficient to apply a few defined rules. To integrate elementary functions, however there is no easy and not all cases a concealing algorithm. Here you are much more likely to rely on numerical approximations. Numerical integration is also called **quadrature**.

This is the simplest form of an approximation. The function graph is approximated using one reference point (n=1) and the area is approximates by a rectangle. At the composite rectangle rule (n>1) the integration region is divided and the above rule is applied to each sub-interval.

Here the basic idea is to approximate the function graph by a straight line by means of two reference points and thus to approximate the area by a trapezoid. At the composite trapezoidal rule (n>1) the integration region is divided and the above rule is applied to each sub-interval. In this function evaluations can be reused, since each inner reference point belongs to two trapezoids.

Here the function is approximated by a quadratic polynomial. Both edges and the center are used as reference points. At the composite Simpson's rule (n>1) the integration region is divided and the above rule is applied to each sub-interval. Again function evaluations can be reused, as the edge nodes usually belong to two intervals.