Midpoint Rule

The Midpoint rule helps estimate the area under a curve by dividing the interval into equal subintervals.

The midpoint of each subinterval is calculated as the average of its two endpoints. The function’s value at these midpoints gives the height of rectangles drawn over them. This approach often gives a more accurate estimate than methods using left or right endpoints, as it reduces errors caused by underestimating or overestimating the area.

Consider a region beside a river. It is bounded by two vertical lines on the sides, a horizontal line below, and a curved riverbank above.

To estimate its area, the region is divided into several rectangles of equal width.

The height of each rectangle is found using the function at the midpoint of its base.

The area of each rectangle is found by multiplying its height by the subinterval width. These areas are added to approximate the total area under the curve, giving the Midpoint rule formula.

As the number of rectangles increases and they become narrower, the estimate improves and approaches the actual area.