Maths IA · Calculus · topic 10 of 40
Area under a curve: trapezium or Simpson? · IB Maths IA idea: guide, sample graph and step-by-step plan
easy to start, easy to do wellTwo methods, one area. Which is closer, by how much, and what does it cost? The evaluation practically writes itself.
- ✓ you like comparing methods and asking which is better
- ✓ you want data that means something to you
- ✓ you like a clear error analysis
The big idea
the trend is the interesting bitMany real quantities are areas under a curve: distance from speed, energy from power, total rainfall from a rate. When you only have data points, you approximate the area. The trapezium rule joins points with straight lines, Simpson's rule with parabolas. Simpson's rule is in neither course, so derive it yourself before you use it. How much better is the second, and does it always win? To answer that you need a reference value, a way to measure error, and a look at how the error changes as you use more points. That is where the depth is.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Solar panel power over a day, a phone's acceleration, a heart-rate trace, or a river's flow.
→ The area has a meaning (energy, distance, volume), so your conclusion is about the real quantity, not just about numbers.Use a function whose integral you can do exactly, or a very fine measurement, as the 'truth'.
→ You can state the error of each method as a number, and plot it. Without a reference you can only compare methods with each other.Repeat with 2, 4, 8, 16 intervals and see how the error falls.
→ Your graph becomes error against step size, and you can estimate how fast each method improves.Real data is rarely equally spaced. Adapt the methods to unequal intervals.
→ Simpson's rule in its standard form no longer applies. You adapt it, or say why you cannot.Add the midpoint rule, or Romberg's method, which builds on the trapezium rule.
→ More to compare, and a chance to explain why combining two approximations can give a better one.Real data has noise. Estimate how much that noise changes the area, compared with the method's own error.
→ You separate two kinds of error and say which one limits your answer.Your Maths IA research question: run the check
Write your own question first. Then make sure it ticks every box:
- Names the data or function, and what the area represents.
- Names the methods you will compare (and the reference value you will compare them with).
- Says how error will be measured: absolute, percentage, or both.
- Says what you will vary to see a trend, such as the number of intervals.
- Is narrow enough for 12 to 20 pages and shows how the methods work, not just their results.
The picture to draw
The same data points, joined two ways. Draw your own with your real points, and shade the area each method adds up.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Your data, why you chose it, and what the area stands for. A prediction about which method will be closer.
Each rule derived or explained from the shapes, with your own notation. One worked example by hand for each.
Where it came from, how it was cleaned, a labelled table or a summary, and the reference value with how you got it.
Results for each method in a table, with the number of intervals, the area and the error.
Error against step size on a graph, what trend it shows, and how it compares with what the theory says.
Which method is better for your data, by how much, and when would the answer change?
Noise in the data, the reference value, smoothness of the curve, the cost of more points. At least one comment in the body too.
Sources for the methods and the data.
What your Maths IA graph could look like
Made-up form: errors for a smooth test function, not for your data. Yours will look different. example only
- Error is plotted against step size, so the reader can see how each method improves as you add points.
- Both methods share axes. The size of the gap is the message.
- To plot error you need a reference value. Say where yours came from and how reliable it is.
- With real data, noise will stop the error from shrinking forever. Say where that happens.
- Another graph to consider: both axes on a log scale. If the points fall on a straight line, its gradient tells you how fast the method improves.
Compare the methods at one step size
one step size only!One step size, three methods, one number each. A chart like this with your own data makes the comparison clear at a glance, but it only tells you about that step size.
Your game plan
get a reference value firstTick steps off as you go. We'll remember on this device.
0 of 10 done
- Choose the dataPick data where the area means something. Check you can say what it represents and in what units.
- PredictBefore computing, say which method you expect to win and by roughly how much. Say what that is based on.
- Learn both rules properlyDerive each from the shapes, work an example by hand, and check it against software.
- Trial run (1 lesson)Try the methods on a function you can integrate exactly. Do they behave as you expected?
- Get your reference valueDecide what counts as the truth for your data, and how sure you are of it.
- Apply the methodsCompute the area by each method at several step sizes, keeping a clean table.
- Study the errorPlot error against step size. What pattern do you see, and does it match what you learned about each rule?
- Add noise and limitsCheck what measurement noise does, and where more points stop helping.
- Reflect as you goAfter each result, say what it means for choosing a method in practice. HL: say what is proved and what is only observed.
- Cut, check, then sendCheck axes, units, notation (no calculator syntax) and that the conclusion answers your aim. Give your teacher one draft.
Maths you'll need
- Integration (AA HL) →
- Approximation and error (AI HL) →
- Integration (AI HL) →
- How the Maths IA is marked →
Traps to avoid in this Maths IA
- Comparing the methods only with each other: with no reference value you cannot say which is closer to the truth.
- One step size only. A trend needs several.
- Concluding Simpson's is 'better' when noise in the data is larger than either method's error.
- Using Simpson's rule with an odd number of intervals.
- Copying the rules from a website without ever showing where they come from.
- Screenshots of spreadsheets filling whole pages. Show one worked example and a results table.
- At HL: only applying the rules. Say something about why they work or how the error is bounded.
More Maths IA ideas like this
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- Vase or bowl: how close can integration get?●●○
This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.