The regular interval approaches all derive from the trapezoid rule through the powerful concept of Richardson extrapolation (Section 4). We therefore need consider in detail only the theory of the trapezoid rule.
Figure 1 illustrates the basic concept underlying regular-interval approaches, decomposition of the desired integral
into a sum of integrals over N regular intervals of width
where 
for 
. The rationale for this
analysis is that the narrowness of the intervals allows the
development of expansions for the 
. For the trapezoid rule,
the 
become N+1 sample points for the integration.
Figure 1: Decomposition of the integral 
into N
regular intervals
Narrowness of the intervals i ensures that Taylor series
about the center points 
well
represent f(x) over the corresponding interval,
where 
, and we define the value and derivatives
of the function at the midpoint as 
. The integral over interval i is then
where all of the odd n terms integrate to zero by symmetry.
Figure 2: Integration of interval .
From (6), the lowest-order approximation to is
the so-called midpoint rule. (See Figure 2.) The associated error is second order,
where 
denotes that the remaining terms are of the order
or higher.
If the are our sample points, we have no direct knowledge of the
, however. Working with the sample points nearest to 
,
and 
, we do have
knowledge of the values
where we have used the series (5). Multiplying the
average of these two values with h gives the trapezoid rule for
approximating ,
This also has a second-order error,
