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.
Surveyors often calculate the area of uneven land strips, like the region between a straight road and a curved riverbank.
Since the river’s curved boundary usually does not have a simple mathematical equation, Simpson’s Rule is used.
Simpson’s rule, based on a numerical integration method, estimates the area using only a few sampled measurements.
Here, these samples are perpendicular distances from the road to the riverbank, measured at three equally spaced positions.
Simpson's Rule works by mathematically fitting a unique parabola through these three points.
The area under the parabola is then found by integrating the quadratic equation over this interval.
After solving the integral and applying the limits, the total area is obtained in terms of h and constants.
The center point allows us to find the constant term, C, immediately. Substituting C into the outer point expressions and solving these expressions reduces it to an expression in terms of y0, y1, and y2.
Finally, plugging these into our integration result yields the formula for the estimated area of the irregular land strip.
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