Simpson's Rule I

Simpson’s Rule is a numerical integration method used to approximate the value of a definite integral when an exact antiderivative is difficult or impossible to obtain. The method estimates area by fitting a unique parabola through three equally spaced points on a curve and then integrating the resulting quadratic function over the interval. By using only a small number of sampled values, Simpson’s Rule provides an accurate approximation for many smoothly varying functions.

A common practical application of Simpson’s Rule arises in surveying, where the area of irregular land must often be calculated. Surveyors may need to determine the area of a strip of land between a straight road and a curved riverbank. Because the riverbank typically does not follow a simple mathematical equation, direct analytical integration is not feasible. Instead, perpendicular distances from the road to the riverbank are measured at three equally spaced positions along the strip.

These measured distances serve as the data points used to construct the approximating parabola. The midpoint measurement allows the constant term of the quadratic function to be determined directly. Substituting this value into the expressions for the remaining points reduces the equation to one involving only the measured distances, commonly denoted as y₀​, y₁​, and y₂​. Integrating the quadratic function and applying the limits yields a formula for the estimated area in terms of the spacing between measurements and the sampled values, providing a practical method for estimating areas of uneven land.