# Rectangle Rule

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Rectangle Rule

a simple formula for the approximate evaluation of a definite integral. It has the form

where *h* = (*b* -*a*)/*n*,*x _{k}* =

*ξ*+ (

*k*- 1)

*h*, and

*a*≤

*ξ*≤

*a*+

*h*.

The most accurate form of the rectangle rule is the midpoint rule, in which *ξ* = *a* + *h*/2. If ǀ*f*″ (*x*)ǀ < *M* on the interval *[a, b]*, this formula satisfies the inequality

In general, the other forms of the rectangle rule are less accurate. Therefore, in place of formulas in which *ξ* = *a* and *ξ* = *a* + *h*, it is preferable to use the arithmetic means of these formulas, since the error will then be at most (*b* — α)^{2}*M* /1 2*n*^{2}. This method is called the trapezoid rule. If both sides of the rectangle rules for *ξ* = *a* + *h*/2, *ξ* = *a*, and *ξ* = *a* + *h* are multiplied by 2/3, 1/6, and 1/6, respectively, and then added together, a more accurate formula for approximate integration is obtained. This formula is called Simpson’s rule. The error in this case is at most (*b* — a)^{5}*N*/2,880n^{4}, where *N* is the maximum value of ǀ*f*^{(4)}(*x*)ǀ on the interval *[a, b]*.