It is not possible to numerically integrate the orbit of Halley more than a few tens of apparitions.

To numerically integrate a function of a two-dimensional vector, equally spaced grid points over a two-dimensional surface are required.

Soon you may be able to find a free-ware computer program to help you numerically integrate in 3-D the attractions and/or repulsions of the space-charge-densities of the electron clouds.

Finite element analysis can be used to numerically integrate often by breaking the flight into small time increments.

Then, the actual angle of the distant star, is determined by numerically integrating from to infinity.

N-body problems can be solved by numerically integrating the differential equations of motion.

It is widely used, for example, by naval architects to numerically integrate hull offsets and cross-sectional areas to determine volumes and centroids of ships or lifeboats.

Beeman's algorithm is a method for numerically integrating ordinary differential equations of order 2, more specifically Newton's equations of motion .

All previous attempts to numerically integrate in supercomputers the Hilbert-Einstein equations describing the gravitational field around binary black holes led to software failure before a single orbit was completed.

Instead, the calculations are performed by any of a variety of computer programs that solve for the fields for non-simple cavity shapes, and then numerically integrate the above expressions.