Правило средней точки для двойных интегралов

In a civil engineering project, a road is planned through mountainous terrain. To estimate the earthwork required, engineers calculate the volume of soil between the existing terrain and the planned road level.

The planned path of the road is modeled as a rectangular region in the xy-plane. A continuous height function represents the varying elevation of the terrain above this base.

To estimate the volume of soil to be excavated or filled, the region is divided into smaller, equal sections. The area of each section is then calculated by the product of its sides.

The midpoint rule approximates a double integral by evaluating the function at each sub-rectangle's midpoint. Within each section, a central point is identified to estimate the approximate height.

The soil height is evaluated at each midpoint. These values approximate the height difference within each section.

From each section, a vertical column of soil is formed by multiplying the rectangular area of each section by its corresponding height.

Similarly, approximate volumes for each column are calculated. Adding all these small volumes gives an approximate total volume of soil above the reference level.