How to use the trapezium rule on the Casio fx-CG50, TI-84 Plus CE and TI-Nspire CX II
On the fx-CG50: MENU → 7 (TABLE) → F5 (SET).On the TI-84 Plus CE: 2nd window (TBLSET).On the TI-Nspire CX II: menu → Table → Edit Table Settings.
Get the y-values from a table of values, put them into the trapezium rule without rounding, and check the answer against the calculator's integral. Exact keystrokes for the fx-CG50 first, then the other calculators, a worked example you can check your answers against, the mistakes that cost marks, and the same thing in Desmos.
When you need it
The trapezoidal rule is in AI (SL and HL): estimate an area from a table of values or a function, and say whether the estimate is too big or too small. AA does not include it.
Practise it: SL AI integration · HL AI integration and kinematics.
Which IB course and paper
| Course | GDC papers | What you use it for |
|---|---|---|
| AA SL | Not in the syllabus | — |
| AA HL | Not in the syllabus | — |
| AI SL | Papers 1 and 2 | The trapezoidal rule from a table of values or a function, and whether the estimate is too big or too small. |
| AI HL | Papers 1, 2 and 3 | As SL. |
A GDC is allowed on every AI paper and on AA Papers 2 and 3; AA Paper 1 is calculator-free, so learn the method by hand as well.
Topic notes: Integration and area under a curve (AI SL) · Integration (AI HL).
Keystrokes on your calculator
Pick your calculator: the page remembers it next time.
Trapezium rule on the Casio fx-CG50
Written for OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell). Casio fx-CG50: graphic (GDC).
- MENU 7 (TABLE). Y1 = √(X³ + 1) EXE.
- F5 (SET): Start 0 EXE, End 2 EXE, Step 0.5 EXE, EXIT, then F6 (TABLE).
- MENU 1 (RUN-MATRIX): 0.5 ÷ 2 × (1 + 3 + 2 × (1.0606601… + 1.4142135… + 2.0916500…)) EXE.
- Check with the integral: see Numerical derivatives and integrals for the ∫dx steps.
Trapezium rule on the TI-84 Plus CE
Written for OS 5.8.x. TI-84 Plus CE: graphic (GDC); also the TI-84 Plus CE Python / CE-T Python Edition.
- y=: Y1 = √(X³ + 1).
- 2nd window (TBLSET): TblStart = 0, ΔTbl = 0.5. Then 2nd graph (TABLE) for the y-values.
- Home screen: 0.5 ÷ 2 × (1 + 3 + 2 × (1.0606601… + 1.4142135… + 2.0916500…)) enter.
- Check: math 9: fnInt( (the ∫ template): limits 0 and 2, the function √(X³ + 1) and X after d, enter.
Trapezium rule on the TI-Nspire CX II
Written for OS 6.x. TI-Nspire CX II: graphic (GDC), non-CAS; the CX II CAS is not allowed in IB, A Level or IGCSE exams.
- Graphs page: f1(x) = √(x³ + 1). ctrl T shows the table.
- menu → Table → Edit Table Settings: Table Start 0, Table Step 0.5.
- Calculator page: 0.5 ÷ 2 × (f1(0) + f1(2) + 2(f1(0.5) + f1(1) + f1(1.5))) uses the function directly, so nothing is rounded.
- Check: the integral template ∫ from 0 to 2 of f1(x) dx.
For teachers: Show on the board Print a keystroke card Teaching GDC skills
Worked example
Use the trapezium rule with 4 strips to estimate ∫₀² √(1 + x³) dx. Is it an overestimate or an underestimate?
Answers (checked with Python)
h = (2 − 0) ÷ 4 = 0.5, so x = 0, 0.5, 1, 1.5, 2 and y = 1, 1.06066, 1.41421, 2.09165, 3. Estimate = 0.5 ÷ 2 × [1 + 3 + 2(1.06066 + 1.41421 + 2.09165)] = 3.28326. The integral is 3.24131, so the trapezium rule overestimates here: the curve bends upwards (it is convex) on this interval, so each trapezium sits above it.
Common calculator mistakes
- Strips and ordinates: 4 strips need 5 y-values (x = 0, 0.5, 1, 1.5, 2). Using 4 y-values gives the wrong h and the wrong sum.
- h is the strip width (b − a) ÷ n, not the number of strips.
- Doubling the first and last y-values too: only the middle ones are doubled.
- Rounding each y-value to 2 or 3 decimal places before adding: the estimate drifts. Keep at least 4 decimal places, or use the full values.
- Over or under: a curve that bends upwards (convex) gives an overestimate; one that bends downwards (concave) gives an underestimate.
- Accuracy: unless the question says otherwise, IB markschemes want answers exact or to 3 significant figures. Keep the full calculator value for later parts and round only the answer you write down (x = 2.094551… is written 2.09, not 2.1 or 2.0946).
If you get an error
fx-CG50: Ma ERROR
Ma ERROR Casio's Math ERROR: the result is outside the allowable calculation range, or you are trying to do an illegal mathematical operation, such as dividing by zero. Check: A function that is undefined somewhere between the limits, such as ln x or 1/x at x = 0, or a limit typed into the wrong box. Fix: Casio's advice: check your input values and change them so the result is within the allowable range.
TI-84 Plus CE: ERR:DOMAIN, ERR:INVALID DIM, ERR:SYNTAX
ERR:DOMAIN A value given to a function or instruction is outside the values it accepts. In a regression, TI says it means the data do not suit the model you chose. Check: A function that is undefined somewhere between the limits, such as ln x or 1/x at x = 0, or a limit typed into the wrong box. Fix: Change the value that is out of range, or choose a model that suits the data.
ERR:INVALID DIM TI's knowledge base: one or more stat plots are turned on, but the lists they use are empty. Check: A stat plot left on from a statistics question, so the error appears when you graph a function. Fix: Turn off the stat plots you are not using, or put the data back in the lists.
ERR:SYNTAX The command contains a syntax error: a misplaced function, argument, bracket or comma. Check: TI points out that subtract − and negative (−) are different keys, and using the wrong one causes this error. Then check the brackets and commas. Fix: Find the misplaced function, argument, bracket or comma and correct it.
TI-Nspire CX II: Domain error, Syntax error
Domain error An argument must be in a specified domain, and this one is not (TI's example: rand(0)). Check: A function that is undefined somewhere between the limits, such as ln x or 1/x at x = 0, or a limit typed into the wrong box. Fix: Change the value that is out of range.
Syntax error The syntax checker found a problem. Check: A missing bracket or comma. Fix: Correct the expression and try again.
Do it in Desmos
Not in the exam. Desmos is not an approved calculator for IB exams, so never rely on it for Paper 2. It is excellent for learning, checking a GDC answer and building graphs for your Internal Assessment (IA).
- Define f(x) = √(1 + x³).
- Type the trapezium rule as a sum: 0.25(f(0) + f(2) + 2(f(0.5) + f(1) + f(1.5))).
- Compare with ∫ from 0 to 2 of f(x) dx (type int or use the keypad).
Open desmos.com/calculator (free, in any browser) and type:
f(x)=√{1+x^{3}}0.25(f(0)+f(2)+2(f(0.5)+f(1)+f(1.5)))int_{0}^{2}f(x)dx
Questions
Can I use a calculator for trapezium rule in the exam?
In IB Maths you can use one approved GDC on every AI paper and on AA Papers 2 and 3, in exam mode. AA Paper 1 is calculator-free, so learn the method by hand as well.
What are the fx-CG50 keystrokes for trapezium rule?
MENU 7 (TABLE). Y1 = √(X³ + 1) EXE. Then F5 (SET): Start 0 EXE, End 2 EXE, Step 0.5 EXE, EXIT, then F6 (TABLE). Then MENU 1 (RUN-MATRIX): 0.5 ÷ 2 × (1 + 3 + 2 × (1.0606601… + 1.4142135… + 2.0916500…)) EXE. The full steps and the other calculators are in the keystrokes section above.
Does this work on other calculators?
Yes. This page has keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire CX II. The worked example has the same answers on every model, so use it to check.
What should I write as working?
Write the maths, not the key presses: say what you are finding in normal notation (the equation, the probability statement or the integral), then the calculator's answer exact or to 3 significant figures unless the question says otherwise. Keep full accuracy in later parts.
More calculator skills
What else is free here
- Every syllabus topicA page for each AA and AI sub-topic, with exam-style questions and mark schemes.
- GDC keystrokes by topicTI-Nspire and Casio fx-CG50 guides for the calculator papers.
- Casio fx-CG50 guideEvery menu you need for IB Maths and the exam traps, on one page.
- Free formula sheets (PDF)One page per course, marking which formulas are not in the IB booklet.
Save your scores and practise the questions that use thisA free account remembers the questions you have done and brings you back to this page.
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Sources
- Casio fx-CG50 Software User's Guide, OS 3.70 (PDF) : Casio fx-CG50, OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell).
- TI-84 Plus CE guidebook (Texas Instruments) : TI-84 Plus CE, OS 5.8.x.
- TI-Nspire CX II guidebooks (Texas Instruments) : TI-Nspire CX II, OS 6.x.
- Desmos Help Center for the Desmos functions.
Menu names are taken from each manufacturer's manual for the models shown. Menus can move slightly between OS updates: if a screen looks different on yours, the manual for your version is the reference, and please tell us so we can update this page. A calculator with no tab here has no checked steps for this skill yet. Worked-example answers were recomputed independently in Python.