### One-dimensional Integration

Numerical integration is a method to approximate a definite integral up to a given degree of accuracy.

where are integration weights and are integration points. It is also called Gauss point. In the following table, you can find Gauss-Legendre quadrature points and weights.

Thus, if the two-point integration rule is used, then

For example, if , then

The exact solution is

For higher order functions, you may need to use more integration points, see the Gaussian Quadrature page on Wikipedia for more information.

### Two-dimensional Integration

For two-dimensional domain , we usually use the product rule with integration points, thus

The integration points and weights are given in the following table.

This integration scheme is often used in the calculation of stiffness matrix for four-node quadrilateral elements.

### Three-dimensional Integration

For a three-dimensional domain such as,

we usually use the product rule with integration points, thus

The integration points and weights are given in the following table.

This integration scheme is often used in the calculation of stiffness matrix for eight-node hexahedral elements.