Numerical Integration

Based on the strategy of replacing a complicated function or tabulated data with an approximating function that is easy to integrate.

Numerical quadrature

The basic method involved in approximating:

Degree of Accuracy

Aka. precision. The largest positive integer such that the quadrature formula is exact for for all .

Trapezoidal Rule

Equivalent to approximating the trapezoidal area under the straight line connecting and .

Degree of accuracy is .

Simpson’s Rules

A more accurate method of approximating the integral of a function, is to use higher-order polynomials to connect the points.

Degree of accuracy is .

Simpson’s 1/3 Rule

Resulted when a second-order interpolating polynomial with 3 equally-spaced points is used to approximate the integral.

Here is between and and . Preferred method because, third-order accuracy is attained with 3 points.

Simpson’s 3/8 Rule

Resulted when a third-order interpolating polynomial with 4 equally-spaced points is used to approximate the integral.

Here is between and and .

Composite Numerical Integration

Composite Trapezoidal Rule

An improved version of the trapezoidal rule. Uses multiple intervals (say ) to approximate the integral.

Here and is between and .

Composite Simpson’s Rule

An improved version of Simpson’s 1/3 rule. Uses even number of multiple intervals to approximate the integral.

Here and is between and .