IB Maths: The Trapezoidal Rule (AI) — Explained

The trapezoidal rule approximates the area under a curve by splitting it into equal-width strips, treating each as a trapezium, and summing their areas. In IB Maths AI (SL and HL) it's examined in Paper 1 (no GDC) as a short formula-substitution question and in Paper 2 with tables of values or a GDC-generated model.

The formula is ${\int }_a^{b}ydx\approx \frac{h}{2}\left(y_0+2(y_1+\cdots +y_{n-1})+y_n\right)$, where h is the strip width. Find h using $h=\frac{b-a}{n}$, list your y-values from the function or table, double all the middle ones, add the two end ones, then multiply by h over 2.

It's an AI-only topic — Analysis and Approaches students never study it. If your child is choosing between AA and AI, this is one small but real content difference: AI leans toward numerical and applied methods like this rule, while AA stays with exact analytic integration techniques throughout.

It depends on the curve's shape. For a concave-up (convex) curve, the trapezoidal rule overestimates the true area, because straight-line tops sit above the curve. For a concave-down curve, it underestimates. Examiners sometimes ask you to state and justify which, so knowing the shape matters as much as the calculation.

Forgetting to double the middle y-values, or miscounting strips versus data points — with n strips you always have n+1 y-values, not n. I've marked hundreds of scripts where a student lists five y-values but treats it as five strips instead of four, throwing off the whole answer.

Use exactly what the question specifies — IB questions always state the number of strips or give you a table with a fixed number of values, so there's no judgement call to make. More strips generally means a closer approximation to the true integral, but on exam papers the number is fixed for you.

Show your h calculation, list every y-value clearly (labelled y0 to yn), substitute carefully into the formula, and state your final answer to the required decimal places. Markschemes award method marks for the substitution step even if your arithmetic slips, so never skip straight to a final number.

No — it's one of the more mechanical, learnable topics in AI, worth roughly 3-5 marks when it appears. The difficulty isn't the concept itself but careless setup errors: wrong h, miscounted values, forgetting to double the middle terms. Students who practise the formula methodically tend to secure these marks reliably.

For Paper 1 you calculate by hand, so no GDC skills are needed there. For Paper 2, your GDC can generate y-values from a function (using a table mode) and check your final area, but IB still expects you to write out the formula and substitution — a bare numerical answer loses method marks.

Integration gives the exact area under a curve using antiderivatives; the trapezoidal rule gives an approximation using straight-line segments, which is useful when you only have data points or a function that's hard to integrate directly. IB AI tests the trapezoidal rule specifically because it models real, messy, non-analytic data.

The rule itself is identical at SL and HL — same formula, same method. What changes at HL is the context: questions are more likely to combine it with harder functions, larger data sets, or ask for justified accuracy comments alongside other HL-only content like more complex numerical methods.

Targeted Topical Worksheets that isolate this one skill work faster than working through a whole textbook chapter, since the rule itself only takes an hour or two to master with the right practice questions. On RevisionPrep, students can find AI-specific Revision Notes and worksheets that pair the formula with exam-style mark-scheme practice.