Area of Irregular Region with Unequal Intervals

Simpson one‐third formula for calculating the area of an irregular region requires that the intervals between measurements be equal. In practice, however, taking measurements at equal intervals may be neither economical, especially if the region boundary exhibits varying fluctuations, nor physically possible. This paper develops a generalization of Simpson one‐third formula that allows the use of unequal intervals. The generalized Simpson (GS) formula is first derived for three ordinates based on Lagrange interpolating polynomial. A composite GS formula for calculating the area given any even number of intervals is then presented. An extended formula is developed for cases in which the number of intervals is odd. The extended formula gives the area for the last interval based on a quadratic polynomial which approximates the boundary shape. An algorithm for calculating the area (using even or odd unequal intervals) is presented. Application of the formulas is illustrated by a numerical example.