Computation by Simpson’s 1/3 rule (double ordinates method): If d is the uniform interval between consecutive ordinates h1, h2, …, hn along a baseline, the approximate area under the boundary by Simpson’s rule is

Introduction / Context:
Area computation from offset ordinates is a staple of surveying. Simpson’s 1/3 rule improves accuracy by fitting parabolas through successive pairs of intervals. Knowing the correct weight pattern on even and odd interior ordinates avoids large numerical errors in earthwork or plotting calculations.

Given Data / Assumptions:

• Ordinates h1 to hn are measured at equal spacing d along a straight baseline.
• Number of intervals (n − 1) is even, as required by Simpson’s 1/3 rule.
• Boundary is sufficiently smooth for parabolic approximation to be reasonable.

Concept / Approach:
Simpson’s 1/3 rule assigns weights 1 to the first and last ordinates, weight 4 to even-numbered interior ordinates, and weight 2 to odd-numbered interior ordinates. The composite area is then multiplied by d/3. This contrasts with the trapezoidal rule, which uses weights 1, 2, …, 2, 1 and a factor d/2, and with crude averaging methods not suited to curved boundaries.

Step-by-Step Solution:

Verification / Alternative check:
For a quadratic boundary y = ax^2 + bx + c, Simpson’s rule gives the exact area, which confirms the weight pattern and the d/3 multiplier.

Why Other Options Are Wrong:

• (d / 2) formula is the trapezoidal rule, not Simpson’s.
• Average-of-ordinates times d is a rough method, typically inaccurate for curved lines.
• Option D swaps the 4 and 2 weights, which is incorrect for Simpson’s 1/3 rule.

Common Pitfalls:
Using Simpson’s rule with an odd number of intervals; mixing up which interior ordinates are “even” and “odd” relative to h1; forgetting the d/3 factor.

Final Answer:
Area = (d / 3) * [ h1 + hn + 4(h2 + h4 + … + h_{n-1}) + 2(h3 + h5 + … + h_{n-2}) ]