Q11. Volume by the Trapezoidal Formula Method is determined by the formula:
📚 Detailed Explanation: The Correct Trapezoidal Volume Formula
The trapezoidal rule for computing volumes from cross-sections follows directly from the elementary trapezoidal rule for areas. When n+1 cross-sections are surveyed at equal spacing D, the total volume is the sum of individual trapezoidal slabs, each of thickness D and whose two faces are consecutive cross-sections.
V = D × [(A₀ + Aₙ)/2 + A₁ + A₂ + A₃ + … + Aₙ⁻₁]
The first (A₀) and last (Aₙ) cross-sections contribute half each. All intermediate sections contribute fully once.
Derivation by Summing Trapezoidal Slabs
Vâ‚â‚‚ = D × (Aâ‚ + Aâ‚‚)/2
V₂₃ = D × (A₂ + A₃)/2
â‹®
V(n-1)n = D × (A(n-1) + Aₙ)/2
Total V = D × [Aâ‚€/2 + Aâ‚ + Aâ‚‚ + … + A(n-1) + Aâ‚™/2]
= D × [(Aâ‚€ + Aâ‚™)/2 + Aâ‚ + Aâ‚‚ + … + A(n-1)] ✓ Option D
Why the Other Options Are Wrong
A: Halves A₀ and Aₙ correctly, but then only sums even-indexed intermediates (A₂, A₄, …) — omitting A₁, A₃, etc. This is the Simpson’s rule coefficient pattern applied incorrectly to the trapezoidal formula.
B: Halves A₁ and Aₙ instead of A₀ and Aₙ. The first section is indexed 0 in the zero-based notation (or 1 in one-based notation) — this option incorrectly uses A₁ as one endpoint, leaving A₀ as a middle section with coefficient 1, which overestimates its contribution.
C: Halves A₀ and A₁ together (the first two sections) instead of A₀ and Aₙ (first and last). This entirely misidentifies which sections are the endpoints and which are intermediate.
Pattern Comparison: Trapezoidal vs Simpson’s
| Section Position | Trapezoidal Coefficient | Simpson’s Coefficient |
|---|---|---|
| First (A₀) and Last (Aₙ) | ½ | 1 |
| Even-indexed intermediate (A₂, A₄, …) | 1 | 4 |
| Odd-indexed intermediate (A₁, A₃, …) | 1 | 2 |
| Overall multiplier | D | D/3 |
Key Concepts for Students
- Memory aid for trapezoidal coefficients: “Half the ends, full for the rest.” Every section except the first and last gets coefficient 1; the first and last each get ½. Multiply the entire sum by D.
- Zero-based vs one-based indexing: Some textbooks start at A₁ (one-based), others at A₀ (zero-based). The formula structure is identical — only the subscript of the first and last section changes. Always identify which is the first section and which is the last before substituting.
- Number of intervals vs sections: n intervals require n+1 sections. For 4 sections (A₀ to A₃), there are 3 intervals: V = D[(A₀+A₃)/2 + A₁ + A₂]. Confusing sections and intervals is a frequent error in these formula questions.