6.1: Elementary Formulas

We first consider integration from 0 to , with small, to serve as the building blocks for integration over larger domains. We here define as the following integral:

To perform this integral, we consider a Taylor series expansion of about the value :

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 and , and simplifying the integral depending on whether the integrand is even or odd, we have

The integrals that we need here are

Therefore,

6.1.2. Trapezoidal rule

From the Taylor series expansion of about , we have

and

Adding and multiplying by we obtain

We now substitute for the first term on the right-hand-side using the midpoint rule formula:

and solving for , we find

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 . Multiplying (6.3) by two and adding to (6.4), we obtain

or

Usually, Simpson’s rule is written by considering the three consecutive points and . Substituting , we obtain the standard result