Deep Dive: The Trapezium Rule

Provenance (2026-09-29): written after reading research/veridian/WMA12-verified-facts.md in full (§1–§9, particularly §2's spec transcription, §3's formula-booklet page, and §6.9/§7/§8's trapezium-rule material and its own two audit passes), content/veridian/curriculum/wma12/the-trapezium-rule.ts in full (897 lines, every block), and all three existing WMA12 technique-hub documents (codex-formula-and-method-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md), plus a grep of content/veridian/question-bank/wma12.ts for trapezium content (none found — nothing to cross-check there). All arithmetic in Part 2 below is original to this document, computed in Python and cross-checked by hand before being written in; none of it reproduces a real exam question's numbers.

#CODEX Deep-Dive: The Trapezium Rule (WMA12, spec 8.3)

#Scope, held deliberately

This document covers exactly spec item 8.3 — approximation of the area under a curve using the trapezium rule — and no more. It is not spec 8.1/8.2 (evaluation of definite integrals; the definite integral as an area), which is codex-deep-dive-definite-integrals-and-areas.md's own scope, held there for the identical reason this document holds 8.3: the lesson files themselves split the two spec items deliberately (the-trapezium-rule.ts's own header names its prerequisite file drawing exactly this line).

A genuine point of confusion worth naming once, explicitly, because the facts bank's own §6.9 groups both under one heading. "Trapezium" appears in this course's research in two unrelated senses. Spec 8.3 (this document) is a numerical method: approximate an area you may not be able to integrate exactly, using a chain of straight-edged trapezia. Spec 8.2 (the sibling document) separately has a real confirmed anchor, June 2025 Q3, where the region bounded by a curve and a line happens to be shaped like a trapezium — an ordinary geometry fact about that particular region, with three creditable strategies for finding its area, none of which is "the trapezium rule." That anchor belongs entirely to the sibling document's Part 3 table and is not repeated here.

What already exists and is not repeated here. The lesson file (the-trapezium-rule.ts) already derives the compressed formula term-by-term from first principles; already carries the topic's single richest real anchor (October 2023 Q6) worked through in a full B1/M1/A1 mark-scheme skeleton; already teaches the graph-reading input format with a purpose-built diagram and chain-drill; already carries five trap-taxonomy items sourced to four named series; and already states the concavity-based reasoning mechanism for the over/underestimate mark in enough depth that the CODEX formula bank and FORGE guide both point back to it rather than re-deriving it. SIGNAL already gives the topic's confirmed frequency (6 of 20 series checked) and its per-series citation list. None of that is reproduced below. What this document adds: two sub-cases the spec explicitly permits but the lesson's own coverage of is thin (a numeric, two-strip-count comparison illustrating "an estimate of the error"; the rule embedded inside another spec topic's own machinery rather than tested in isolation); one structural sub-case the lesson never drills at all (a multi-part question where an earlier part's trapezium-rule result must be reused, not recomputed, in a later part); and a single synthesis table of every real exam angle this specific spec item has actually been tested from, gathered from across the facts bank and SIGNAL rather than sitting piecemeal in either.

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#Part 1 — The real model, compactly

#1.1 The formula, and its given/memorised status

$\int_a^b y\,dx \approx \frac{1}{2}h\{(y_0+y_n) + 2(y_1+y_2+\cdots+y_{n-1})\}$, with $h = \frac{b-a}{n}$ — printed in full on the Pure Mathematics P1/P2 page of the formula booklet, the $h$ formula included. Nothing about the formula's shape needs to be recalled from memory; what the marks are actually for is building the right ordinate table and substituting it into the right places, which is exactly what the-trapezium-rule.ts's own derivation (writing every individual trapezium's area out in full, then collecting the shared interior edges) makes mechanical rather than a rule to trust blindly. Not re-derived here.

#1.2 What the spec text itself actually permits, read closely

Spec 8.3 is short — one content line plus one guidance sentence — and both halves matter independently. The content line names one method: approximating $\int_a^b y\,dx$ with the trapezium rule. The guidance sentence permits two further things a question is allowed to do with that method: ask for more strips to sharpen the estimate, and ask for a sense of how far off an estimate is. In practice, on every real anchor this course has confirmed, "how far off" has meant the qualitative DIRECTION (over/underestimate, argued from the curve's concavity — a one-mark reasoning item, and the lesson's own central worked example). This document found no confirmed real anchor asking for the second, numeric kind explicitly (see Part 4), but the spec's own wording permits it, and Part 2.4 below builds the short worked comparison the lesson's own coverage of that permission — one MCQ, no worked numbers — doesn't carry.

#1.3 The three real input formats, unchanged from the lesson

A question can hand you the ordinates directly (a stated $n$ and a function), as a ready-built table, or — the confirmed harder format — plotted and labelled on a graph with no formula or table given at all, a skill the lesson's own diagram-and-chain-drill pair drills directly. Pointer only; not re-derived.

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#Part 2 — Sub-cases, with short original micro-examples

Three sub-cases below are already fully covered by the lesson and are named, not re-derived. Two are genuinely thin in the existing lesson (one MCQ each, no worked numbers), and one is not drilled there at all — those three get a short original micro-example each.

#2.1 A single trapezium-rule application, correctly structured — pointer only

Fully covered: the lesson's worked chain reproduces the spec's own guidance example ($\int_0^1\sqrt{2x+1}\,dx$, 4 strips) in full, and its first chain-drill repeats the structure with $\int_0^2\frac{1}{x+1}\,dx$. Nothing to add.

#2.2 Reading ordinates from a labelled graph rather than a table or formula — pointer only

Fully covered by the lesson's own diagram (a plotted pond cross-section with two zero-valued endpoint ordinates) and the chain-drill immediately after it, built specifically to close a confirmed real audit gap. Nothing to add.

#2.3 Justifying the over- or underestimate direction from concavity — pointer only

Fully covered, and the lesson's own stated central focus: two mechanism blocks, both chain-drills' final stages, the marked-solution's part (c), and two of the five trap-taxonomy items all attack this one confirmed hardest-mark-on-the-paper gap from different angles. Nothing to add.

#2.4 Increasing the strip count — a short original numeric comparison

The spec permits asking for an increasing number of trapezia to sharpen an estimate, and the lesson's tz-mcq-6 states the qualitative direction (more, narrower strips move the estimate closer to the true value) without ever computing two estimates side by side. Here is that comparison, worked once, cleanly:

$\int_0^2 x^3\,dx$, compared at $n=2$ and $n=4$.

$n=2$: $h=1$, ordinates $y_0=0,\,y_1=1,\,y_2=8$. Estimate $=\frac{1}{2}(1)\{(0+8)+2(1)\}=5$.

$n=4$: $h=0.5$, ordinates $y_0=0,\,y_1=0.125,\,y_2=1,\,y_3=3.375,\,y_4=8$. Estimate $=\frac{1}{2}(0.5)\{(0+8)+2(0.125+1+3.375)\}=0.25\times17=4.25$.

True value (spec 8.1): $\int_0^2 x^3\,dx=\left[\frac{x^4}{4}\right]_0^2=4$.

Both estimates overestimate — $y=x^3$ bends away from the x-axis throughout $[0,2]$, the same bending direction as $y=x^2$ elsewhere in this lesson, so every chord sits above the curve — and doubling the strip count from 2 to 4 shrank the error from $1$ to $0.25$, a quarter of its previous size. That specific ratio is a feature of this one curve on this one interval, checked directly, not a general law this document asserts holds for every curve — but the direction (more strips, smaller error) is exactly what the spec's guidance permits asking about, and this is the worked instance of it this course's materials were missing.

#2.5 The trapezium rule embedded inside another spec topic's own machinery — a short original illustration

The lesson's own fifth trap-taxonomy item names this pattern from two real anchors — a genuinely novel format testing the rule "via logarithms" (January 2025, part (i)) and a real-life water-flow context wrapped around the same mechanism (October 2023 Q6) — but states it only in prose, with no worked example of what "embedded inside" actually looks like arithmetically. This is not a reconstruction of the real January 2025 question — its content beyond the phrase quoted in the facts bank is not established anywhere in this course's sources — but a VERIDIAN-original illustration of the general mechanism the citation names: an extra step, outside the trapezium rule itself, has to run before the ordinate table can even be built.

A curve given by $\log_{10}y = 0.3x+0.1$, approximated on $[0,4]$ with 4 strips. Before any ordinate can be found, the relationship has to be rearranged: $y=10^{0.3x+0.1}$. Only then does the familiar machinery start: $h=1$, ordinates $y_0=1.2589,\,y_1=2.5119,\,y_2=5.0119,\,y_3=10.0000,\,y_4=19.9526$ (4 d.p.), giving an estimate of $\frac{1}{2}(1)\{(1.2589+19.9526)+2(2.5119+5.0119+10)\}\approx28.1296$. (Exact integral, for comparison: $\frac{10^{1.3}-10^{0.1}}{0.3\ln10}\approx27.0619$ — an overestimate, since $y=10^{0.3x+0.1}$ curves upward throughout, the same reasoning as 2.4 above.) The teaching point is the un-logging step itself, not the trapezium arithmetic that follows it unchanged: a student who has only practised this rule against a bare function is one algebraic step short of ready for a question that hides the function inside another relationship first.

#2.6 A multi-part question where an earlier trapezium-rule result must be reused, not recomputed — a short original illustration

Genuinely undrilled anywhere in the lesson, and grounded in two independent real confirmations rather than invented from nothing: the confirmed real structure of October 2023 Q6 (compute an area by the trapezium rule in one part, then use that area for a subsequent real-world calculation in the next) and the confirmed real error of June 2025 Q5(b), where candidates who recomputed the trapezium rule instead of using their part (a) answer scored nothing on that part and lost time besides.

An irrigation channel's cross-section is measured at five equally spaced points, 2 m apart, with depths $0,\,1.2,\,1.6,\,1.2,\,0$ m. (a) Estimate the cross-sectional area. (b) Water flows through the channel at $0.4\text{ m/s}$; using your answer to part (a), estimate the volume of water passing a fixed point each second.

(a) $h=2$, area $\approx\frac{1}{2}(2)\{(0+0)+2(1.2+1.6+1.2)\}=1\times8=8\text{ m}^2$.

(b) Correct route: $8\times0.4=3.2\text{ m}^3\text{s}^{-1}$ — one multiplication, using the value already found. The confirmed real error this illustrates: recomputing part (a)'s trapezium sum a second time inside part (b) — even correctly — earns none of part (b)'s marks, because those marks are for the step part (b) actually asks for (the multiplication), not a repetition of part (a)'s. The cost is not just the mark: it is time spent re-deriving a number already sitting on the page.

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#Part 3 — Synthesis: every real exam angle this topic has actually been tested from

Six series carry a confirmed, individually-cited real trapezium-rule (spec 8.3) question in this course's archive — matching SIGNAL's own "6 of 20" count exactly:

What this table shows that no single existing document states in one place: four of the six anchors are error-taxonomy findings about the same two structural slips (strip-width miscount; endpoint/interior swap or bracket mishandling) recurring across four different series spanning three years (2020–2024) — a degree of repetition this course's archive has not confirmed to the same extent for the sibling spec items 8.1/8.2, whose own three confirmed anchors (per the sibling document's Part 3) are each a distinct, single-instance real question rather than the same error recurring across series. The other two anchors here (January 2025, June 2025) are, independently, both questions that refuse to test the rule in isolation — one folding it inside another spec topic's algebra, the other folding it inside a multi-part structure that punishes recomputing instead of reusing. Read together, the six anchors say the topic has two real faces: an arithmetic-structure face that is heavily, repeatedly tested the same few ways, and a "the rule shows up somewhere unexpected" face that is real but thinly documented in this course's own sources.

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#Part 4 — What remains genuinely unverified

• June 2024's specific question, part structure, and mark-scheme skeleton are not established in this course's archive beyond its inclusion in the facts bank's grouped "at least four series" citation for the strip-width/doubling error taxonomy. Nothing here fills that in with invented specifics.
• January 2025's "trapezium rule via logarithms" question is confirmed to exist and confirmed to be a genuinely novel format, but the facts bank's own source quote is truncated ("Many candidates knew...") and no fuller description of the actual mechanism survives in this course's sources. Part 2.5 above is stated explicitly as a VERIDIAN-original illustration of the general pattern the label names, not a reconstruction of this real question.
• June 2025 Q5's exact part-by-part structure — which part is the trapezium-rule step, what the paired "definite integral" part actually asks for, and whether that pairing is the source of spec 8.3's own error-estimate permission — is a plausible reading of SIGNAL's frequency-table note and FORGE's follow-through-trap quote taken together, but this document did not independently verify it against a primary mark scheme for this series. Flagged as inference, not fact.
• No confirmed real WMA12 anchor asks for a numeric error estimate or a two-strip-count comparison as its own gradable step. Every confirmed anchor testing "how far off" a trapezium-rule estimate is tests the qualitative DIRECTION only (over/underestimate, from concavity). The spec's own wording permits the numeric version (Part 1.2), and Part 2.4 above builds the worked comparison this course's materials were missing, but it remains untested against a real graded question in this archive.
• October 2025 and January 2026 examiner reports were not found in any of the three local archives this course has searched (per the facts bank's own §8) — commentary from those two series, if it exists, has not yet been checked for this topic.

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Sources used, by name: research/veridian/WMA12-verified-facts.md §2 (spec transcription), §3 (formula-booklet page), §6.9/§7/§8 (trapezium-rule findings, frequency, and flagged gaps); content/veridian/curriculum/wma12/the-trapezium-rule.ts (full file); content/veridian/technique-hub/wma12/codex-formula-and-method-bank.md, forge-exam-technique-guide.md, and signal-frequency-and-boundaries.md (all read in full, for cross-checking and to avoid restating what they already state). No real WMA12 exam question is reproduced in this document; all worked numbers in Part 2 are VERIDIAN-original and independently computed before being written in.