Mathematics 11–12 · Year 12

Volume of a flask, a horizontal pipe and a bottle neck, checked with water

Further integration: integration by substitution, integration of rational functions by partial fraction decomposition, and trigonometric products written as sums and differences (Mathematics Extension 2, Year 12)

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The idea

The measured profile of a vessel decides the form of the integral that gives its volume: a straight taper needs a linear substitution, a circular cross-section needs a trigonometric substitution and the double-angle identity, and a hyperbolic taper needs a rational integrand, and water in a measuring cylinder settles whether the integration was right.

What you need

A 250 mL conical flask, or another straight-sided tapering vessel, and vernier callipers reading to 0.1 mm
A 500 mm length of 90 mm PVC stormwater pipe with two push-on end caps sealed with silicone by the teacher before the lesson, and a 15 mm hole drilled in the top of the pipe wall at mid-length for filling and for the dipstick; the internal diameter measured with the callipers (84 mm on the pipe used here, so R = 42.0 mm)
A plastic drink bottle with a tapering neck, cut off below the shoulder by the teacher and the cut edge taped
A 500 mL measuring cylinder reading to 5 mL and a 10 mL graduated cylinder reading to 0.1 mL
A steel rule marked in millimetres and narrow enough to pass through the filling hole, as a dipstick, and a waterproof marker
A spirit level, a large tray or a sink to work over, a funnel, and food dye to make the water level easy to read
A spreadsheet for the profile fits and the depth-volume table

How to do it

Measure the conical flask's outside diameter at the base of the cone and at the top of the cone with the callipers and halve each, and measure the vertical height between them (radii 42.5 mm and 17.0 mm and height 95 mm on the flask used here).
Write the radius as a linear function of height, integrate pi r^2 by the substitution u = a + b x, and state the volume in millilitres using 1 mL = 1000 mm^3.
Fill the flask to the top of the cone from the 500 mL measuring cylinder, recording the volume added over three fills, and compare the mean with the integral and with the frustum formula pi h (R^2 + R r + r^2) / 3.
Lay the sealed pipe horizontally and level it with the spirit level. Derive the wetted cross-sectional area at depth d by integrating 2 sqrt(R^2 - x^2), using the substitution x = R sin(theta) and then cos^2(theta) = (1 + cos(2 theta)) / 2.
Add water through the filling hole in 100 mL steps, reading the depth with the dipstick held square to the surface each time, for at least 20 readings, and tabulate depth against volume.
Compare every measured pair with V(d) = L (R^2 arccos((R - d) / R) - (R - d) sqrt(2 R d - d^2)), and plot both the measured points and the curve.
Mark a calibration scale on the dipstick from the formula, then test it: have a partner add an unrecorded volume and predict it from the depth alone.
Measure the bottle neck's outside diameter at three heights 10 mm apart with the callipers, subtract twice the wall thickness measured at the cut edge, and halve the result (inside radii 13.0 mm, 11.0 mm and 9.5 mm on the bottle used here). Fit r = p / (h + q) from the first two readings and test the fit against the third.
Integrate pi p^2 / (h + q)^2 over the 20 mm of neck, a rational integrand with a repeated linear factor, and also compute the volume of a cone frustum with the same end radii.
Fill the cut neck from the 10 mL graduated cylinder three times, take the mean, and state which of the two models the measurement supports.

What you should see

The flask's cone with R = 42.5 mm, r = 17.0 mm and h = 95 mm integrates to 280.3 mL, and the frustum formula returns the same value because the two expressions are identical once the substitution is made; the water measurement comes out below that, because radii read on the outside of the glass include its wall and the base of a real flask is rounded where the frustum has a sharp corner, and the shortfall as a percentage of 280.3 mL is the learner's measure of how far those two effects reach. The pipe with R = 42.0 mm and L = 500 mm holds 2771 mL full and 1385 mL at half depth; at d = 21.0 mm, a quarter of the diameter, it holds 542 mL, which is 19.6 per cent of the full volume, and at d = 10.0 mm it holds 186 mL, 6.7 per cent of the full volume at 11.9 per cent of the depth. The measured depth-volume pairs track the curve to within the error of reading the depth, which is far larger than the 5 mL resolution of the cylinder: one millimetre of depth holds the pipe length times the width of the water surface, 27 mL at 10.0 mm, 36 mL at 21.0 mm and 42 mL at half depth, where the surface is the full 84 mm wide. The plot is visibly not a straight line. The neck radii 13.0 mm and 11.0 mm measured 10 mm apart give p = 715 mm^2 and q = 55 mm, and that fit predicts 9.53 mm at the third height against a measured 9.5 mm; the rational integral pi p^2 (1 / q - 1 / (q + 20)) gives 7.79 mL, while a cone between the same end radii gives 8.04 mL, and the 0.25 mL difference is above the 0.1 mL resolution of the small cylinder. The learner knows it worked when the measured pipe volumes sit within the depth-reading error of the curve, when the mean neck measurement lies clearly nearer one of the two neck models than the other, and when the dipstick scale predicts an unrecorded addition to the pipe to within the volume that one millimetre of depth holds at that level.

What changes

What you change: depth of water in the horizontal pipe, and height along each measured profile
What you measure: volume held
What you keep the same: the pipe levelled before every run
the same measuring cylinder and the same reading technique
the dipstick held square to the water surface
water at room temperature
the same vessel drained and wiped between fills

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

The volume held is proportional to the depth; the pipe at a quarter of its diameter holds 19.6 per cent of its contents, not 25 per cent, and only at exactly half depth, by symmetry, does it hold half.
An average radius can be squared and multiplied by the height; squaring an average is not the same as integrating the square.
Every taper can be treated as a cone; the hyperbolic fit for the neck gives 7.79 mL where a cone with the same end radii gives 8.04 mL.
The technique is a matter of preference; the measured profile fixes the integrand, and the integrand fixes whether a substitution, a rational decomposition or a double-angle identity is needed.
Water measures the volume exactly; the wall thickness, the meniscus and the film left behind all shift the reading, which is why every prediction comes with a stated tolerance or an expected shortfall.
A definite integral needs the antiderivative in the original variable; a substitution may carry the limits with it instead.

Safety card

Low riskLearners carry it out

Hazards

glassware breaking
water spilled on the floor or near a power point
sharp edges on the cut pipe and the cut bottle

Controls

work over a tray or in a sink and carry the flask with two hands
wipe spills at once and keep laptops and leads off the bench in use
have the teacher cut and cap the pipe and the bottle beforehand and tape the cut edges

Note

Water and food-grade dye only, with no chemicals and no heat, so the NSW Department of Education Chemical Safety in Schools package does not apply; the glassware and the cut edges belong in the school's risk assessment before the lesson.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Extension 2 11–12 Syllabus (2024), Year 12 focus area Further integration; Extension 2 is a Year 12 course taught from Term 4 2026 with the first HSC examination in 2027, and it replaces the Mathematics Extension 2 Stage 6 Syllabus (2017) taught until then; page read 2026-09-23ME2-12-04
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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