INTEGRATION

The trapezoidal rule improves upon the zeroth-order rule by replacing the constant with a linear representation of the function, f(s). This is accomplished by including another, higher-order term, in the polynomial 5↑. Since this is a “closed” integration with specific integration limits from A to B, the independent variable, s is redefined so that it is explicitly valid over the interval of integration. Using the definition of a derivative over a known interval

when this function is replaced into the integral approximation, eqn. 4↑, the integration yields

with an error of

where s ∈ (A, B). This is often referred to as the midpoint rule.

A modification or “correction” to the trapezoidal rule which can improve predictions is

While the truncation error and integrated value may be more accurate, this version of the trapezoidal rule requires that the derivatives of the points also be known or can be estimated.

Keeping another term (n = 2) in the polynomial approximation for f_n in eqn. 5↑ results in a series of schemes known as Simpson’s rules. The basic Simpson rule yields an integrated equation

with a truncation error of

where s ∈ (A, B)

When equal spacing is available, these formulas can be combined to offer significant computational savings. The trapezoidal rule for example becomes

for the entire length B − A with a truncation error of

and the Simpson basic (1/3) rule becomes

and a truncation error of