There are many situations which call for the replacement of an integral by a sum, or vice versa. In the former case the highest accuracy is sometimes required. This means that in the expression

the choice of weights w_i and knowledge of the error terms are crucial. Thankfully this extreme concern with detail is not necessary for us. We assume that the region under the curve may be replaced by a sequence of contiguous rectangles on equal bases. (Strictly speaking only half the area of the end rectangles should be included in the sum if the trapezoidal or midpoint rules are followed.)