Rectangle Rule

Figure 7.2: The approximate area under the curve f(x), using rectangles whose height is such that their left upper point lies on the curve f(x).

A first, very simple, scheme is to approximate the function f(x) on each interval by a constant, and considering the area of the rectangle of width h = x_i+1 - x_i and height f(x_i) for the i^th interval. It is then a simple matter to compute the sum of the areas of all the rectangles from $x = \alpha$ to $x = \beta$. The choice of using rectangles whose height agrees with the function f(x) at their left sides introduces a bias in the estimate of the integral. We would be equally justified in choosing the height of each rectangle so that the right upper corner lies on the curve f(x), or even such that the mid-point of the top of each rectangle lies on the curve f(x). If we choose a sufficiently small step-size, h, we can attain sufficient accuracy in the value of the integral, but at the cost of many evaluations of the integrand, f(x).

It is clear that not choosing a sufficiently small step-size can lead to a systematic error. Consider the integral of the function, f(x), shown in Figure 7.2, but now for x in the interval from x₁ to the value of x where the maximum of f(x) occurs. Each of the rectangles in this restricted range of x lies completely the curve f(x). The area of each rectangle is less than the area under that segment of the curve f(x) by the area of what is roughly a triangle of base length h and height f(x_i+1) - f(x_i).

If we now consider the integral from the maximum of f(x) to x_n+1, a similar effect is apparent, but now the rectangles yield areas that are in excess of the area under f(x) by the areas of roughly triangles with similarly defined areas.