Estimating Pit Excavation Volume Using Unequal Intervals

The current methods of estimating the excavation volume for a rectangular pit use excavation depth measurements at lattice points of a grid formed by marking off each of the x‐ and y‐axes in intervals of equal length. The excavation volume is estimated by using the trapezoidal rule or Simpson's 1/3 and 3/8 formulas along with these depth measurements to estimate the double integral of the excavation depth function. In this paper we first derive a generalized Simpson (GS) 3/8 formula to estimate the integral of a function over an interval partitioned into three possibly unequal intervals. We assume the function is approximately a cubic with known ordinates at the four points that partition the interval. We then assume that the excavation depth measurements of the rectangular pit are taken as the lattice points of a grid in which the axes are partitioned into intervals of unequal length and use the GS 3/8 formula with the known GS 1/3 formula to derive the excavation volume estimate. A numerical example is included.