Earthwork by sections at equal spacing: identify the correct Trapezoidal Rule for volume An embankment is divided into n cross-sections at equal spacing D (metres). If A1, A2, …, An are the respective end-area cross-sectional areas (m^2), which expression correctly gives the Trapezoidal Rule volume V?
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AV = (D/2) * [A1 + An + 2*(A2 + A3 + ... + A_{n-1})]
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BV = (D/3) * [A1 + An + 4*(A2 + A4 + ... ) + 2*(A3 + A5 + ... )]
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CV = (D/3) * [A1 + An + 2*(A2 + A4 + ... + A_{n-1}) + 4*(A3 + A5 + ... + A_{n-2})]
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DV = D * (A1 + An)/2 (Average end-area only; ignores interior areas)
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EV = D * (A1 + A2 + ... + An) (simple summation without weights)
Answer
Correct Answer: V = (D/2) * [A1 + An + 2*(A2 + A3 + ... + A_{n-1})]
Explanation
Introduction / Context:For linear earthworks such as roads and canals, volumes are commonly computed from a sequence of cross-sections spaced at equal distances. Two widely used numerical methods are the Trapezoidal Rule and Simpson’s Rule. Correct identification of each formula avoids systematic bias in estimated quantities and costs.
Given Data / Assumptions:
- n sections at equal spacing D.
- Areas A1 (first) to An (last) represent cross-sections perpendicular to the centerline.
- Ground varies smoothly between sections.
Concept / Approach:
The Trapezoidal Rule treats each adjacent pair of sections as forming a prismoid whose end areas are averaged, then multiplied by spacing D. Summing across all intervals leads to the compact weighted sum with coefficients 1 for the ends and 2 for all interior areas. Simpson’s Rule, in contrast, uses alternating 4 and 2 coefficients and requires an odd number of segments (even number of intervals) with D/3 as the multiplier.
Step-by-Step Solution:
Start with one interval: V_i ≈ D * (A_i + A_{i+1})/2.Sum over i = 1 to n−1: V = Σ V_i = (D/2) * [A1 + An + 2*(A2 + ... + A_{n-1})].Hence, select the expression with D/2 and weight 2 on all interior areas.Verification / Alternative check:
Dimensional check: Areas in m^2 times D in m gives m^3. Weighting matches the trapezoidal composite integration rule.
Why Other Options Are Wrong:
Option B/C are Simpson-type formulas (require D/3 and alternating 4,2 weights), not trapezoidal.
Option D ignores interior areas beyond the ends and is incomplete.
Option E lacks proper weighting and overcounts interior sections.
Common Pitfalls:
Confusing trapezoidal and Simpson weights; using D instead of D/2; forgetting to include all interior areas with coefficient 2.
Final Answer:
V = (D/2) * [A1 + An + 2*(A2 + A3 + ... + A_{n-1})]