The Midpoint Rule for Double Integrals

The midpoint rule divides a rectangular region into smaller sub-rectangles of equal size. Within each sub-rectangle, the function is evaluated at the midpoint to determine the approximate height. The volume of each vertical column equals the height at the midpoint multiplied by the sub-rectangle's area. Summing all column volumes gives the total approximate volume over the region.

In civil engineering, the midpoint rule estimates soil volume between terrain and a planned road level. The road footprint is modeled as a rectangle, and terrain elevation is represented by a continuous height function. By partitioning the region and evaluating height at each midpoint, engineers can approximate excavation or fill requirements without computing exact integrals, making it practical for real-life applications of multiple integrals.

Each sub-rectangle has an area ΔA = Δx · Δy, where Δx and Δy are the widths in the x and y directions. The volume of soil in each column is calculated by multiplying the height at the midpoint, f(xi*, yj*), by this area. The total approximate volume is the sum of all such products across all sub-rectangles in the partition.

Dividing the rectangular base into m intervals in the x-direction and n intervals in the y-direction creates smaller, more manageable sections. As the region is divided into finer sub-rectangles, the midpoint height becomes a more accurate representative value for each section. This refinement increases the accuracy of the total volume approximation.

The continuous height function f(x,y) represents the terrain elevation above a reference level at any point in the rectangular region. At each sub-rectangle's midpoint (xi*, yj*), the function value f(xi*, yj*) serves as the approximate height for that entire section. This function value, multiplied by the sub-rectangle area, estimates the soil volume in that column.

The total approximate volume is obtained by summing the volumes of all vertical columns: the double sum of f(xi*, yj*) · ΔA over all sub-rectangles. Each term represents one column's volume, calculated by multiplying the midpoint height by the sub-rectangle area. Adding these contributions across the entire partitioned region yields the overall soil volume estimate.

Accuracy depends on the fineness of the partition. As sub-rectangles become smaller, the midpoint height more closely represents the average height across each section. Finer partitions with more sub-rectangles produce better approximations because the height variation within each smaller section is reduced, making the midpoint value more representative of the entire section.