It is sometimes useful to organize the computations for an application of Simpson's Rule in a table. Here is a table showing the computations of the last example. Relating The Trapezoidal, Midpoint, and Simpson's estimates. M. Thus . It appears that when a function is concave up, the definite integral lies between the midpoint and trapezoid estimates.
Similar analysis works for a function that is concave down on an interval, leading to .
Again the definite integral lies between the midpoint and trapezoid estimates. So as long as we have a function that can be analyzed in sections where the concavity is well determined to be the same we have that the definite integral will be between the trapezoidal and midpoint estimates. When the concavity is not the same throughout the interval there is still some relation between these three estimates. So from the algebra we have that S N is between T k and M k without reference to the concavity of f.