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A double integral gives the volume under a surface defined by a function of two variables over its domain, shown as the region in the plane.

Consider an excavation site defined by a rectangular region on the xy-plane. Above this region lies a surface whose height represents the ground elevation.

The total volume of soil to be excavated is the space between this surface and the rectangular base.

To calculate this volume, the base is divided into a grid of small subrectangles, each with an area ΔA.

For each subrectangle, a sample point is chosen, and the function value at that point gives the height.

Multiplying this height by ΔA forms a vertical rectangular soil column.

Adding the volumes of all the columns across the grid gives an approximate total volume. This approximation is called a double Riemann sum. As the number of subrectangles increases within a fixed region, the approximation improves.

Taking the limit as the grid becomes infinitely fine, and as each subrectangle’s area approaches zero, defines the double integral.

The double integral represents the exact volume beneath the surface over the region.