Midpoint Rule

Approximating areas under curved boundaries is a common problem in applied mathematics, particularly when an exact calculation is difficult or impractical. One effective numerical method for this purpose is the Midpoint Rule, which provides an estimate of the area under a curve by using rectangular approximations over a specified interval.

Description of the Midpoint Rule

The Midpoint Rule begins by dividing the given interval into a number of equal subintervals. For each subinterval, the midpoint is identified by averaging the values at its two endpoints. The value of the function at this midpoint is then used to determine the height of a rectangle constructed over the subinterval. Because the midpoint often lies closer to the average behavior of the function on that interval, this method tends to reduce the error associated with overestimation or underestimation that can occur when left or right endpoints are used.

Geometric Interpretation

A practical example of the Midpoint Rule can be illustrated by estimating the area of a region adjacent to a river. The region is bounded on the left and right by vertical lines, below by a horizontal line, and above by a curved riverbank. Since the upper boundary is irregular, its area cannot be easily computed using simple geometric formulas.

Area Approximation Process

To approximate the area of this region, it is divided into several rectangles of equal width. The height of each rectangle is determined by evaluating the boundary curve at the midpoint of the rectangle’s base. The area of each rectangle is calculated by multiplying its height by its width. Adding the areas of all rectangles produces an estimate of the total area under the curve, which is the essence of the Midpoint Rule.

Accuracy of the Method

As the number of rectangles increases and their widths decrease, the rectangular approximation more closely follows the shape of the curve. Consequently, the estimated area improves and approaches the true area of the region.