Simpson's Rule I

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.