5.12 Areas and volumes questions for IB Maths AI

36 questions on Areas and volumes, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 5 Calculus
HL only

Questions by paper and level

Paper	Questions
Paper 1	18
Paper 2	17
Paper 3	1
Every paper	36

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-01171Paper 1 · 6 marks

Consider the function $f (x) = x^{3} - 5 x^{2} + 4 x$ , for $0 \leq x \leq 4$ . Part of the graph of $y = f (x)$ is shown below.

y = f(x)

The region $R$ is enclosed by the graph of $f$ and the $x$ -axis for $0 \leq x \leq 4$ .

(a)
Write down the $x$ -coordinate of the point where the graph of $f$ crosses the $x$ -axis for $0 < x < 4$ .
1 mark
(b)
Find the area of $R$ .
3 marks
(c)
Rhys says that the area of $R$ is equal to $\int_{0}^{4} f (x) d x$ . Explain why Rhys is incorrect.
2 marks

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Question 2 · AI-00777Paper 2 · 14 marks

A glassmaker in Murano designs a vase. Its shape is formed by rotating the region between the graph of

f (x) = 4 + 2 \sin (0.4 x), 0 \leq x \leq 15,

and the $x$ -axis through $2 π$ radians about the $x$ -axis. All lengths are in cm. Ignore the thickness of the glass.

(a)
Find the maximum radius of the vase and the value of $x$ at which it occurs.
2 marks
(b)
Write down an integral for the volume of the vase, and find this volume.
3 marks
(c)
A second vase is designed by applying to the graph of $f$ a horizontal stretch with scale factor 1.5 and a vertical stretch with scale factor 0.8, to give the graph of $y = g (x)$ . Find an expression for $g (x)$ and state its domain.
3 marks
(d)
Find the volume of the second vase.
2 marks
(e)
Show that the ratio of the volume of the second vase to the volume of the first vase is equal to $1.5 \times {0.8}^{2}$ .
2 marks
(f)
A third vase is formed from the graph of $f$ by a horizontal stretch with scale factor 1.5 and a vertical stretch with scale factor $k$ . It has the same volume as the first vase. Find the value of $k$ .
2 marks

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Question 3 · AI-01771Paper 3 · 27 marks

This question investigates the capacity of a glass decanter designed from a fitted curve.

Lucía is a glassmaker. She designs a decanter whose inside is formed by rotating the curve $r = f (x)$ , $0 \leq x \leq 28$ , through $2 π$ radians about the $x$ -axis, where $x$ cm is the height above the base and $r$ cm is the internal radius at that height.

She chooses $f (x) = a x^{3} + b x^{2} + c x + d$ , so that the curve passes through the points $(0, 5)$ , $(8, 8)$ , $(20, 3)$ and $(28, 3.5)$ , as shown in the following graph.

r = f(x)
r = −f(x)

The inner profile of the decanter, lying on its side. The base is at x = 0 and the opening at x = 28.

(a)
Write down the value of $d$ and a system of three equations satisfied by $a$ , $b$ and $c$ .
3 marks
(b)
Find the values of $a$ , $b$ and $c$ .
2 marks
(c)
Use calculus to find the maximum internal radius of the decanter and the height at which it occurs.
3 marks
(d)
Find the internal radius at the narrowest point of the neck.
2 marks
(e)
Find the capacity of the decanter, in litres.
3 marks
(f)(i)
Before fitting the curve, Lucía measured the internal radius of a prototype every 4 cm.
$x$ (cm)
0
4
8
12
16
20
24
28
$r$ (cm)
5.0
7.7
8.0
6.7
4.8
3.0
2.3
3.5
Use the trapezoidal rule with the cross-sectional areas $π r^{2}$ at these heights to estimate the capacity of the prototype, in cm³.
3 marks
(f)(ii)
Find the percentage difference between your answers to parts (e) and (f)(i), taking the answer to part (e) as the true value.
2 marks
(g)
The decanter stands upright and 1 litre of wine is poured into it.
Find the depth of the wine.
3 marks
(h)
Wine is poured into the empty decanter at a constant rate of $50 {cm}^{3} s^{- 1}$ .
Find the rate at which the depth of the wine is increasing when the depth is 10 cm.
3 marks
(i)
Lucía also makes a larger version of the decanter in which every length is multiplied by 1.2.
Find the capacity of the larger decanter, in litres.
2 marks
(j)
Give one reason why the capacity found in part (e) may be different from the volume of wine the decanter actually holds.
1 mark

$x$ (cm)	0	4	8	12	16	20	24	28
$r$ (cm)	5.0	7.7	8.0	6.7	4.8	3.0	2.3	3.5

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Question 4 · AI-01172Paper 1 · 7 marks

The nose cone of a model rocket is formed by rotating the curve $y = 2 \sqrt{x}$ , for $0 \leq x \leq 9$ , through $360 °$ about the $x$ -axis, where $x$ and $y$ are measured in centimetres.

(a)
Find the volume of the nose cone.
3 marks
(b)
Write down the radius of the base of the nose cone.
1 mark
(c)
A designer considers replacing the nose cone with a right circular cone of the same length and the same base radius. Find the percentage by which the volume of the original nose cone exceeds the volume of this cone.
3 marks

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Question 5 · AI-01454Paper 2 · 16 marks

A water tank on a farm is 4 m tall. Its inside is formed by rotating part of the curve

x = 1.2 + 0.4 \ln (y + 1), 0 \leq y \leq 4,

through $2 π$ radians about the $y$ -axis, where $x$ is the internal radius of the tank, in metres, at a height of $y$ metres above its base.

The following table shows some values of $x$ , correct to three decimal places.

$y$ (m)	$0$	$1$	$2$	$3$	$4$
$x$ (m)	$1.200$	$1.477$		$1.755$

(a)
Complete the table.
2 marks
(b)
Use the trapezoidal rule with four intervals to estimate the volume of the tank.
4 marks
(c)(i)
Write down an integral that gives the exact volume of the tank.
2 marks
(c)(ii)
Find the value of this integral.
1 mark
(d)
Calculate the percentage error in the estimate from part (b).
2 marks
(e)
The empty tank is filled at a constant rate of $0.15 m^{3}$ per minute. Find the time taken to fill the tank. Give your answer to the nearest minute.
2 marks
(f)
Find the depth of the water when the tank contains $25 m^{3}$ of water.
3 marks

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Question 6 · AI-01173Paper 1 · 7 marks

The inside of a drinking glass is modelled by rotating the curve $y = 0.4 x^{2}$ , for $0 \leq y \leq 12$ , through $360 °$ about the $y$ -axis. All lengths are in centimetres.

(a)
Express $x^{2}$ in terms of $y$ .
1 mark
(b)
Find the volume of liquid the glass can hold.
3 marks
(c)
Mariana pours $250 {cm}^{3}$ of juice into the empty glass. Find the depth of the juice.
3 marks

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