How to use the trapezium rule on a Casio ClassWiz or fx-CG50
On the fx-991EX: MENU → 9 (Table).On the fx-991CW: HOME → Table.
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 Casio ClassWiz first (the fx-991EX and fx-991CW have different menus, so each has its own tab), then the graphic 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
Numerical integration: the trapezium rule estimate, whether it is an over- or underestimate, and how to improve it. The table of values does the y-values; the formula is still yours to show.
Practise it: Integration · Numerical methods.
Keystrokes on your calculator
Pick your calculator: the page remembers it next time.
Trapezium rule on the Casio fx-991EX ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991EX ClassWiz: scientific (no graphs).
- MENU 9 (Table): f(x) = √(X³ + 1), = (and = again past g(x) if it appears).
- Start 0, End 2, Step 0.5, =: the five y-values appear.
- MENU 1: work out 0.5 ÷ 2 × (1 + 3 + 2 × (1.0606601… + 1.4142135… + 2.0916500…)) using the full values from the table.
- Check: ∫▭ √(X³ + 1) from 0 to 2, =, gives the true value to compare with.
Trapezium rule on the Casio fx-991CW ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991CW ClassWiz: scientific (no graphs); the newer ClassWiz with a HOME menu of apps.
- HOME → Table. Define f(x) = √(x³ + 1), then Table Range: Start 0, End 2, Step 0.5, and EXE.
- Show the table: the five y-values.
- HOME → Calculate: 0.5 ÷ 2 × (1 + 3 + 2 × (sum of the three middle y-values)), EXE.
- Check: CATALOG → Function Analysis → Integration(∫) with √(x³ + 1) from 0 to 2.
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.
If you get an error
fx-991EX: Math ERROR, Syntax ERROR, Time Out
Math ERROR A result, or a value you typed, is outside the range the calculator can handle, or you asked for something mathematically impossible, 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 the values you typed, use fewer digits and try again. If a variable is the input to a function, make sure its value is one the function accepts.
Syntax ERROR There is a problem with the format of the calculation. In Matrix and Vector calculations it also appears when the sizes do not suit the operation. Check: A missing or extra bracket, or two operation signs in a row. With matrices or vectors: adding a 2 × 2 to a 3 × 3, or a product whose sizes do not match. Fix: Casio's advice: make the corrections. For matrices and vectors, check the size you set for each one.
Time Out A derivative or integral calculation ended without meeting its ending condition. Check: An integral of a function that changes very quickly, or over a very wide interval. Fix: Casio's advice: increase the tol (tolerance) value. The answer is then less precise.
fx-991CW: Math ERROR, Syntax ERROR
Math ERROR A result, or a value you typed, is outside the range the calculator can handle, or you asked for something mathematically impossible, 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 the values you typed, use fewer digits and try again. If a variable is the input to a function, make sure its value is one the function accepts.
Syntax ERROR There is a problem with the format of the calculation: it is typed in a form the calculator cannot read. Check: A missing or extra bracket, two operation signs in a row, or a function with nothing after it. Fix: Casio's advice: make the corrections and try again.
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 allowed in A Level exams. Use it to learn, to check a calculator answer, and to see what a function or data set looks like.
- 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?
Yes. Every UK A Level Maths paper (Edexcel 9MA0, AQA 7357, OCR H240) and every Edexcel IAL unit, P1 included, allows a calculator that has no CAS and no wireless link. The fx-CG50 qualifies.
What are the fx-991EX keystrokes for trapezium rule?
MENU 9 (Table): f(x) = √(X³ + 1), = (and = again past g(x) if it appears). Then Start 0, End 2, Step 0.5, =: the five y-values appear. Then MENU 1: work out 0.5 ÷ 2 × (1 + 3 + 2 × (1.0606601… + 1.4142135… + 2.0916500…)) using the full values from the table. 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-991EX ClassWiz, Casio fx-991CW ClassWiz, 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 to the accuracy the question asks for (often 3 significant figures). Keep full accuracy in later parts.
More calculator skills
What else is free here
- A Level Maths topicsEach topic explained, with a worked example and practice questions.
- Past papers and mark schemesLinks to the boards' own papers, with a timer and topic practice.
- IAL revision notesOne page per topic across P1 to P4, S1 and M1.
- Formula sheets (PDF)Edexcel, AQA, OCR and IAL on one page each, showing what each board's booklet gives you.
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Sources
- Casio fx-991EX / fx-570EX Quick Start Guides : Casio fx-991EX ClassWiz, the fixed ROM (these models take no OS updates).
- Casio fx-991CW / fx-570CW User's Guide : Casio fx-991CW ClassWiz, the fixed ROM (these models take no OS updates).
- 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.