The Newton-Cotes formulas (open and closed) that are derived in this section allow us to approximate an integral by a weighted sum of discrete points on [a,b], the interval of interest:
In this section we focus on a single elemental ``unit'', using just two or three points; in the next section we'll paste the units together to get an accurate integral for an interval by pasting the elemental units together, hence using many points.
Our preliminary schemes are based on integrating interpolating polynomials.
-
degree of accuracy (precision) of a quadrature formula is the largest
degree for which polynomials of that degree are integrated exactly using the formula.
- closed Newton-Cotes formulas use the endpoints in the approximation of the integrals, whereas open Newton-Cotes formulas do not use the endpoints in the approximation of the integrals.
We could start with interpolating constant functions (step-functions), and if we did we would begin by deriving the left- and right- rectangle rules. But we won't! We start with the linear interpolating polynomial, from which we derive the
Trapezoidal rule:
where 
. It is derived by integrating the linear Lagrange
intepolating polynomial:
We can use the weighted mean-value theorem to write the second integral as
One interesting (but unsurprising) conclusion is that the Trapezoidal rule gets the integrals of linear functions right: hence its degree of accuracy (or precision) is given by 1.
Simpson's Rule:
can be derived similarly by integrating the Lagrange interpolating quadratic - but that means that we must use three points for the elemental unit of Simpson's rule. The formula works out to
This error term does not arise from the derivation using the Lagrange
quadratic, however: that derivation gives an order 
error term,
dependent on the third derivative. It can be obtained by integrating the Taylor
polynomial of degree 4 expanded about 
(not even an interpolating
polynomial!), and using the centered difference formula for the first
derivative. This seems curious, but - whatever works!
Astonishingly, Simpson's rule gets cubics right! Now here's one of the
interesting twists: we don't have to use the quadratic geometry to understand
Simpson's rule: all cubic functions interpolating three points will have the
same integral between 
and 
! Is that obvious, in any way?
Example #18, p. 196 (linear algebra wins again!)