Table 6.7.1 lists five common fixed-step methods for the numeric evaluation of the definite integral ${\int }_a^{b}f\left(x\right) \mathrm{dx}$. If $h\=\left(b-a\right)\/n$, the values of the integrand $f\left(x\right)$ at the evenly-spaced nodes $x_k\=a\+k h\,k\=0\,1\,\dots \,n$, are denoted by $f_k$ rather than by $f\left(x_k\right)$. Of course, $x_0\=a$ and $x_n\=b$. The third column provides an upper bound on the absolute value of the error made when evaluating an integral by one of the listed methods. The maximum of the absolute value of the appropriate derivative is taken over $x\in \left[a\,b\right]$.

The Left Endpoint, Right Endpoint, and Midpoint rules are examples of "Rectangular" rules because rectangles are used to approximate the area under the graph of the function being integrated. The Trapezoid rule uses trapezoids, and Simpson's rule approximates the function with a sequence of quadratics and finds the areas under these arcs.

The Rectangular rules evaluate the function at $n$ points, but Simpson's rule (as stated in Table 6.7.1) and the Trapezoid rule evaluate the function at $n\+1$ points.

The traditional statement of Simpson's rule given in Table 6.7.1 requires $n$ to be an even integer. This form of the rule is more easily remembered if described as "$h\/3$ times the sum of the first and last function values plus four times the sum of the odd-indexed values plus two times the sum of the even-indexed values." Maple has both this form and an alternate form in which the function is evaluated at the midpoints of the $n$ subintervals generated by the $n$ nodes. In this form, the function is evaluated at $2 n\+1$ points, so when making comparisons of the accuracy of the various methods, it is important to do so with methods that expend comparable computational energy.