Area by Simpson's Rule
Area under an irregular boundary using parabolic strips.
Formula
Variables
Example
Weighted sum of offsets × h/3 gives the plot area
How it works
Simpson’s Rule estimates area by replacing each pair of adjacent strips with a parabola that passes through three measured ordinates, then integrating that curve. The end ordinates get weight 1, the odd-numbered interior ordinates get weight 4, and the even-numbered interior ordinates get weight 2 because that pattern is exactly what comes out of integrating quadratic fits over equal intervals. It is more accurate than the trapezoidal rule when the boundary is smooth and the offsets are taken at equal spacing.
When to use it
Use this in surveying and geomatics when you need the area of an irregular field, river section, excavation, or contour shape from equally spaced offsets along a baseline. It is especially suitable when the boundary is smooth enough that a parabola is a good local approximation.
Worked example, step by step
Suppose offsets are taken every 10 m across a plot, giving y0 = 0, y1 = 6, y2 = 10, y3 = 12, y4 = 9, y5 = 4, y6 = 0. There are 6 strips, so h = 10 m and the ordinates are equally spaced. Odd-indexed ordinates: y1 + y3 + y5 = 6 + 12 + 4 = 22 Even-indexed ordinates: y2 + y4 = 10 + 9 = 19 Apply Simpson’s Rule: A = (h/3)(y0 + 4Σodd + 2Σeven + y6) A = (10/3)(0 + 4(22) + 2(19) + 0) A = (10/3)(88 + 38) A = (10/3)(126) A = 420 m^2 Final answer: the estimated area is 420 m².
Common mistakes to avoid
- Using unequal strip widths, which invalidates the standard Simpson’s Rule formula
- Forgetting to separate the ordinates into odd and even positions before applying the 4 and 2 multipliers
- Including the end ordinates inside the odd/even sums instead of counting them once only
Did You Know?
Simpson's rule needs an even number of strips and is more accurate than the trapezoidal method.
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