6.1: Elementary Formulas
( \newcommand{\kernel}{\mathrm{null}\,}\)
We first consider integration from 0 to
To perform this integral, we consider a Taylor series expansion of
6.1.1. Midpoint rule
The midpoint rule makes use of only the first term in the Taylor series expansion. Here, we will determine the error in this approximation. Integrating,
Changing variables by letting
The integrals that we need here are
Therefore,
6.1.2. Trapezoidal rule
From the Taylor series expansion of
and
Adding and multiplying by
We now substitute for the first term on the right-hand-side using the midpoint rule formula:
and solving for
6.1.3. Simpson’s rule
To obtain Simpson’s rule, we combine the midpoint and trapezoidal rule to eliminate the error term proportional to
or
Usually, Simpson’s rule is written by considering the three consecutive points