IA idea · Numerical methods & error analysis

Trapezoidal rule or Simpson's rule? Areas under the normal curve

AA SLAA HLAI SL Solid Also in: Calculus, Statistics

Research question

How many strips do the trapezoidal rule and Simpson's rule need to find normal-distribution probabilities to four decimal places, and how does each method's error change when the number of strips doubles?

Adapt it: change the place, the data or the comparison until the question is yours.

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Why it makes a good exploration

The normal curve has no elementary antiderivative, so numerical integration is genuinely needed. Measuring how the error shrinks reveals the 'order' of each method, which you can then explain.

The mathematics you'll need

The trapezoidal rule (SL)
Simpson's rule derived from fitting parabolas (new)
Error tables and ratios when h halves (expect 1/4 and 1/16)
Logarithms to find the order of convergence
Normal distribution probabilities

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Modelling the data, step by step

This idea usually needs a quadratic model. See it worked step by step, with a criterion tip at every step: Quadratic: three points, completing the square, regression.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

No data needed; use a GDC's normal CDF as the reference value and say so.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Set up the integral for a probability you care about.
Apply both rules for increasing numbers of strips.
Tabulate errors and their ratios.
Explain the ratios (derive Simpson's rule; HL: use a series to justify the error).
Reflect on the reference value and on when each method is worth it.

Pitfalls that cost marks

Comparing with a reference value you haven't justified.
Too few rows to see the pattern.
Deriving nothing: quoting Simpson's rule without showing where it comes from.

Showing personal engagement

Choose a probability with meaning to you (a height or an exam score).
Predict the strips needed before calculating.
Find a function where Simpson's rule is exact and explain why.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	The trapezoidal rule; Simpson's rule derived from fitting parabolas (new)
AA HL	Good fit	The trapezoidal rule; Simpson's rule derived from fitting parabolas (new)
AI SL	Good fit	The trapezoidal rule; Simpson's rule derived from fitting parabolas (new)
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Choose the function or equation yourself and predict how each method will behave before you run it. Hunting for the cases that break a method is engagement an examiner can see.

Reflection (D)

Reflect on error: how it changes with step size or iterations, why a method converges slowly or not at all, and how you know your 'exact' comparison value is correct. For this idea, start with: comparing with a reference value you haven't justified — say how it affects your answer.

Use of mathematics (E)

SL: The trapezoidal rule or a simple iteration applied correctly, errors tabulated against step size and explained, with any method outside the syllabus explained step by step.

HL: Convergence analysed rather than observed: an error bound derived with calculus or a series, an order of convergence measured and justified, or Euler's method studied against an exact solution.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Use Richardson extrapolation to improve the trapezoidal results and explain why it works.

Extending it for HL

Derive an error bound with a Maclaurin series or calculus, then show your numerical results follow it.

See a complete IA, marked

Our annotated exemplar Is a hanging chain a parabola? Comparing catenary and quadratic models (AA SL) asks a different question, but shows how a complete numerical methods exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).