Simpson's Rule II

In warehouse roofing applications, corrugated or curved metal sheets are commonly used to improve structural strength, water drainage, and ventilation efficiency. To accurately estimate material requirements and optimize design parameters, engineers must determine the curved surface area of these sheets. Because the sheet profiles often repeat smoothly along their length, they can be effectively approximated by parabolic curves, enabling the use of numerical integration techniques for area estimation.

One widely applied numerical method is Simpson’s Rule, which provides an accurate approximation of the area under a smooth curve. The method begins by selecting two endpoints of the curve, defining the total horizontal interval [a, b]. This interval is divided into n equal subintervals of width

where n must be an even integer. The curve is then sampled at n + 1 equally spaced points x₀, x₁, …, x_n.

Simpson’s Rule operates by grouping every two subintervals, or equivalently every three consecutive points, and approximating the curve over each group with a parabola. The area under each parabolic segment is obtained from an exact integration of the interpolating quadratic function. When applied across the entire interval, this process leads to a characteristic weighted summation of function values. The composite Simpson’s Rule formula is expressed as

The repeating coefficient pattern of 1, 4, and 2 reflects the contributions of endpoints, midpoints, and interior points of each parabolic segment. Summing these weighted terms yields a reliable estimate of the definite integral. For warehouse roofing, this accuracy is particularly valuable, as it supports precise material planning, cost estimation, and structural design decisions while accommodating the curved geometry of the sheets.