Maths IA · Calculus · topic 13 of 40
Vase or bowl: how close can integration get? · IB Maths IA idea: guide, sample graph and step-by-step plan
measure carefully: the maths is the easy partYou can pour water into a vase and read its volume. Can you get the same number from a pencil, a ruler and an integral?
- ✓ you like making a model fit a real object
- ✓ you can measure carefully and keep a table tidy
- ✓ you want to use volumes of revolution properly
The big idea
why this function? that's criterion EMeasure the radius of a vase at many heights, fit a function to that profile, and rotate it about the vertical axis. The volume integral gives a prediction, and a measuring jug gives the truth. The interesting part is the gap. Where does the model lose volume: the fitted curve, the wall thickness, the measuring, the numerical method? At HL the volume of revolution is course content, so criterion E rewards how you choose and justify the profile, not that you can apply the formula once.
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 →
Try a polynomial, a trigonometric form and a spline-like piecewise model for the same profile.
→ You compare models by something better than R², such as the volume each one predicts and where its residuals are largest.Measure the wall thickness and rotate the inner profile instead of the outer one.
→ The integrand changes and the predicted volume drops. You can explain how large the effect is.Use numerical integration directly on measured radii (trapezium, Simpson's).
→ You compare an exact integral of a fitted curve with a numerical one on the raw data, and say which you trust more.Work out volume as a function of fill height, V(h), and mark a scale on the vase.
→ The integral has a variable upper limit. You can check your marks by pouring, which is a strong test.Photograph the vase against a grid and digitise the outline in software.
→ You get far more points, but have to deal with perspective and scale. A good evaluation point.A bowl with an oval rim or a slightly irregular vase.
→ The simple model breaks, so you must decide how to handle the error or switch to slices of a different shape.Your Maths IA research question: run the check
Write your own question first. Then make sure it ticks every box:
- Names the object, and says exactly which volume is being predicted: outer, inner or capacity.
- Says how the profile will be measured (how many heights, what tool) and how it becomes a function.
- States what the prediction will be compared with, such as the measured capacity, and how that is measured.
- Needs a decision that is yours (which function, why this many points), not just one formula applied once.
- Is narrow enough for 12 to 20 pages and uses HL-level calculus with reasons.
The picture to draw
The vase profile with radii marked at equal heights, and one thin slice showing where the formula comes from. Draw yours with your real object and scale.
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 object, why you chose it, and what you expect the model to over- or under-estimate and why.
Where the formula comes from (slices), written in your own words with your own symbols.
How you measured radius and height, how many points and why, with a labelled table and uncertainties.
At least two candidate functions, how each was fitted, and a reason for the final choice beyond 'it fits best'.
The integral set up, worked in full for one model, with a numerical check by hand or software.
A table and graph comparing predicted and poured volume, with a percentage difference and a reason for it.
Back to the aim: how close did integration get, and what limits it?
Rank the sources of the gap (wall, fit, measurement, method). Say which you tested and which you only argued.
Sources for the formula and any software or data.
What your Maths IA graph could look like
Made-up measurements, to show the form of a strong graph, not the answer. Yours will look different. example only
- Data points with error bars, and the fitted curve drawn through them, so the reader judges the fit.
- Axes titled with quantity and units. Note that r is a half-width: say whether it is the outer or inner radius.
- The fit looks good here, but a volume depends on r², so small errors in r are doubled. Say that in the text.
- Residuals matter more than the look: where is the curve consistently above or below the points?
- Another graph to consider: residuals against height, or the cumulative volume V(h) built up slice by slice.
Which choice moves your volume most?
r² doubles the error!Rank by the effect on the volume, not on the raw readings. Because volume depends on r², radius errors count double. Make a chart like this with your own numbers.
Your game plan
pour it twice!Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Choose and predictPick the object. Write down whether you expect the integral to be above or below the poured volume, and why.
- Learn the formula properlyDerive the volume of revolution from slices so you can explain it, then try it on a shape with a known volume, such as a cone.
- Trial run (1 lesson)Measure the radius at a few heights. Is the object symmetric? Where does the base start? Fix your method now.
- Measure the profileTake radius and height at equal steps, with a consistent method. Also measure the wall thickness and record the poured capacity.
- Fit more than one functionFit at least two kinds of function. Decide how you will judge them: residuals, volume, simplicity.
- Integrate by handSet up and evaluate the integral for your chosen model, showing the working. Check with software.
- Compare with the pourPour the water, read the volume, and compare with the prediction as a percentage. Repeat the pour to see its spread.
- Explain the gapTest one cause of the gap (wall, fit, numerical method). Does fixing it close the gap by the amount you expect?
- Reflect as you goNote what each result means for the aim. HL: say why your choice of function is more than convenience.
- Cut, check, then sendCheck every symbol, axis and unit, and that the integral appears in proper notation. Give your teacher one draft.
Maths you'll need
Traps to avoid in this Maths IA
- Choosing the function because it fits best and giving no other reason.
- Measuring the outer radius and then comparing with the capacity, which is the inner volume.
- Forgetting that the volume depends on r squared, so a small radius error gets bigger.
- A profile with too few points. Three or four points cannot support a fancy function.
- Reporting the volume to six figures when the radius is known to the nearest millimetre.
- Numerical integration with no step size, and no check that the answer settles as the step shrinks.
- A high R² quoted as proof of a good model, with no look at the residuals.
More Maths IA ideas like this
- Maclaurin series: how many terms are enough?●●○
- Draining tank: predict the time to empty●●●
- Pursuit curves: does the chaser ever catch up?●●●
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.