The Midpoint Rule for Double Integrals

The midpoint rule for a double integral provides a practical method for estimating volume over a rectangular region when the surface height varies continuously. In civil engineering, this method is useful for approximating the amount of soil to be moved when planning a road across uneven terrain. The road footprint is represented as a rectangle in the xy-plane. At the same time, the terrain elevation above a flat reference level is described by a continuous height function f(x,y). The objective is to approximate the volume of soil lying between the surface and the reference plane.

Partition of the Region

To apply the midpoint rule, the rectangular base region is divided into smaller sub-rectangles of equal size. If the rectangle is partitioned into m intervals in the x-direction and n intervals in the y-direction, then each sub-rectangle has an area

\begin{equation*}\Delta A = \Delta x \, \Delta y\end{equation*}

This subdivision replaces the irregular terrain over the full region with a collection of smaller sections that are easier to analyze.

Midpoint Approximation

Within each sub-rectangle, the midpoint (x_i*, y_j*) is determined. The function value f(x_i*, y_j*) gives the approximate terrain height for that section. Each sub-rectangle is then treated as the base of a vertical column of soil whose volume is estimated by

\begin{equation*}f(x_i^*, y_j^*) \, \Delta A\end{equation*}

Thus, the height at the midpoint serves as a representative value for the entire section.

Estimated Total Volume

The approximate total volume is obtained by summing the volumes of all such columns:

\begin{equation*}\iint_{R} f(x, y)\, dA \approx \sum_{i=1}^{m} \sum_{j=1}^{n} f(x_i^*, y_j^*) \, \Delta A\end{equation*}

This approximation becomes more accurate as the region is divided into finer sub-rectangles, making the midpoint rule an effective tool for estimating earthwork volumes in engineering applications.