Area by ordinates — Simpson’s 1/3 rule for a curvilinear boundary divided into an even number of equal-width strips states that the area equals: Identify the correct coefficient pattern for extreme, odd, and even offsets.

Introduction / Context:
When a boundary is irregular, its area can be estimated from offset ordinates at equal intervals using numerical integration rules. Simpson’s 1/3 rule gives a higher-accuracy estimate than the trapezoidal rule by weighting odd and even ordinates differently. This is a standard technique in surveying computations of areas from field measurements.

Given Data / Assumptions:

• Offsets (ordinates) are taken at equal spacing (strip width = d).
• The number of strips is even (number of ordinates is odd).
• Offsets are ordered from the first extreme to the last extreme.

Concept / Approach:
Simpson’s 1/3 rule formula for area A uses weighting 1 for the two extreme offsets, 4 for odd-numbered interior offsets, and 2 for even-numbered interior offsets. The multiplier outside is d/3. This produces improved accuracy for smooth curves by approximating the boundary with parabolic segments over pairs of strips.

Step-by-Step Solution:

Verification / Alternative check:
Cross-verify by splitting into parabolic pairs and confirming the 1:4:2 pattern across each pair, then summing for the whole boundary.

Why Other Options Are Wrong:

• Options (a), (b), and (d) use incorrect coefficients and/or multipliers; they do not match the 1:4:2 with d/3 rule.
• (e) is wrong because (c) is exactly the Simpson’s 1/3 rule.

Common Pitfalls:
Applying Simpson’s rule with an odd number of strips or confusing which offsets are “odd” vs “even” (count interior positions from the first extreme).

Final Answer:
One-third the strip width * [sum of extreme offsets + 4*(sum of remaining odd) + 2*(sum of even)]