Q12. If d is the constant distance between the sections, then the correct prismoidal formula for volume is:
📚 Detailed Explanation: The Prismoidal (Simpson’s) Volume Formula
The prismoidal formula for volumes is the three-dimensional analogue of Simpson’s rule for areas. It achieves higher accuracy than the trapezoidal method by fitting a parabolic curve through groups of three consecutive cross-sections rather than a straight line through pairs. The formula has a specific and fixed coefficient pattern that must be memorised exactly.
V = (d/3) × [(A₀ + Aₙ) + 4(A₁ + A₃ + …) + 2(A₂ + A₄ + …)]
d = uniform spacing between sections
A₀, Aₙ = first and last section areas (coefficient 1)
Odd-indexed intermediates A₁, A₃, … (coefficient 4)
Even-indexed intermediates A₂, A₄, … (coefficient 2)
Requirement: Number of intervals must be even (odd number of sections).
Why the Coefficients Are 1, 4, 2, 4, 2, …, 1
Simpson’s rule fits a parabola through every group of three consecutive points. Within each group, the integration of the parabola gives weights of 1, 4, 1 to the three points. When groups are chained together, the shared intermediate points accumulate: section at an odd position is shared between two groups as a “centre” point (weight 4), while sections at even positions are shared as “endpoints” of two groups (weights 1+1 = 2).
Why the Other Options Are Wrong
| Option | Error | Consequence |
|---|---|---|
| A | Missing d/3 multiplier (uses plain d); even-area coefficient is 1 instead of 4; odd-area coefficient is 2 instead of its correct role | Drastically underestimates; formula structure is wrong |
| C | Swaps the 4 and 2 coefficients: gives 2 to odd-indexed sections and 4 to even-indexed | Wrong assignment; will give incorrect result for any non-symmetric section sequence |
| D | Uses d/6 instead of d/3 as the multiplier | Halves the correct volume; d/6 is used in the two-section prismoidal formula V = (D/6)(A₁+4Aₑ+A₂), not the multi-section form |
Coefficient Summary for All Volume Methods
| Method | End sections | Odd intermediates | Even intermediates | Multiplier |
|---|---|---|---|---|
| Trapezoidal | 1 | 2 (same as even) | 2 | d/2 |
| Prismoidal (Simpson’s) | 1 | 4 | 2 | d/3 |
Key Concepts for Students
- Memory pattern — 1, 4, 2, 4, 2, …, 4, 1: Start with 1, alternate 4 and 2 for intermediates, end with 1. All multiplied by d/3. If this pattern ends on a 4 (not 2), you’re fine — that means you have an odd number of sections as required.
- Even intervals required: Simpson’s prismoidal rule works in pairs of intervals (3 sections per pair). The total number of intervals must be even. If odd, apply the trapezoidal rule to the last interval and add it to the Simpson result for the rest.
- Single prismoid vs multi-section: V = (D/6)(A₁+4Aₑ+A₂) applies to ONE solid with known top, mid, and bottom areas. V = (d/3)[…1,4,2,4,1…] is the generalised multi-section form. They are different formulas; confusing their multipliers (d/6 vs d/3) is the classic exam trap in Q12-type questions.