5.8 The trapezoidal rule questions for IB Maths AI

31 questions on The trapezoidal rule, 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
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	12	1	13
Paper 2	12	4	16
Paper 3HL only	0	2	2
Every paper	24	7	31

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-01196Paper 1 · 7 marks

The cross-section of a drainage channel is the region enclosed by the curve $y = 2 + 3 x - 0.5 x^{2}$ , the $x$ -axis and the lines $x = 0$ and $x = 6$ , where $x$ and $y$ are measured in metres.

(a)
Use the trapezoidal rule with four intervals of equal width to estimate the area of the cross-section.
3 marks
(b)
Write down an integral for the exact area of the cross-section and find its value.
2 marks
(c)
Calculate the percentage error in the estimate from part (a).
2 marks

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Question 2 · AI-00795Paper 2 · 16 marks

A landscape architect, Priya, designs an earth mound for a park. The cross-section of the mound is modelled by the function

h (x) = - 0.03 x^{3} + 0.36 x^{2} + 1, 0 \leq x \leq 12

where $h (x)$ is the height of the mound, in metres, at a horizontal distance of $x$ metres from one end of the cross-section. The cross-section is the region between the graph of $h$ , the $x$ -axis and the lines $x = 0$ and $x = 12$ .

y = h(x)

(a)
Complete the following table.
$x$
0
2
4
6
8
10
12
$h (x)$
1
2.2
7.48
8.68
1
2 marks
(b)
Use the trapezoidal rule with the values in the table to estimate the area of the cross-section.
3 marks
(c)
Write down an integral for the exact area of the cross-section, and hence find this area.
2 marks
(d)
Calculate the percentage error in the estimate found in part (b).
2 marks
(e)
Priya repeats the estimate using the trapezoidal rule with 12 intervals, and obtains 63.48 m².
Find the percentage error in this estimate, and comment on the effect of using more intervals.
2 marks
(f)
The mound is 30 m long and has the same cross-section along its whole length.
Find the volume of earth needed to build the mound.
2 marks
(g)
Earth costs 18.40 euros (EUR) per m³. Priya's budget for earth is 35 000 EUR.
Determine whether Priya's budget is enough. Justify your answer.
3 marks

$x$	0	2	4	6	8	10	12
$h (x)$	1	2.2		7.48	8.68		1

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Question 3 · AI-01645Paper 3 · HL only · 29 marks

This question uses calculus to design how a tram accelerates and brakes between two stops.

A tram travels in a straight line between two stops, starting and finishing at rest. Its velocity, $v$ m s⁻¹, is recorded every 10 seconds during one journey of 80 seconds.

$t$ (s)	0	10	20	30	40	50	60	70	80
$v$ (m s⁻¹)	0	1.5	4.7	9.1	12.6	15.1	14.2	9.9	0

(a)
Use the trapezoidal rule with all the values in the table to estimate the distance between the two stops.
3 marks
(b)(i)
The tram company's engineer, Mireille, models the velocity by
$v (t) = \frac{t^{2} (80 - t)}{5000}, 0 \leq t \leq 80.$
Find the distance between the stops according to this model.
2 marks
(b)(ii)
Find the percentage difference between your answer to part (a) and your answer to part (b)(i), taking the value in part (b)(i) as the true value.
2 marks
(c)
Use differentiation to find the maximum velocity of the tram according to Mireille's model.
3 marks
(d)
Find the maximum acceleration of the tram according to the model, and the time at which it occurs.
3 marks
(e)
For passenger comfort, the magnitude of the acceleration of a tram must never exceed 1.3 m s⁻².
Show that Mireille's model satisfies this condition.
2 marks
(f)
Find the time at which the tram has travelled half of the distance between the stops.
3 marks
(g)(i)
On a new route, two stops are 900 m apart. Mireille uses a model of the same form,
$v (t) = c t^{2} (T - t), 0 \leq t \leq T,$
where $T$ seconds is the journey time and $c$ is a positive constant.
Show that $c = \frac{10 800}{T^{4}}$ .
3 marks
(g)(ii)
Show that the greatest magnitude of the acceleration during the journey is $\frac{10 800}{T^{2}}$ .
3 marks
(g)(iii)
Hence find the shortest journey time on the new route that satisfies the comfort condition in part (e).
2 marks
(g)(iv)
Find the maximum velocity of the tram on this journey.
2 marks
(h)
State one reason why Mireille's model may not be suitable for the new route.
1 mark

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Question 4 · AI-01438Paper 1 · 5 marks

Ngozi is surveying a small lake in a nature reserve. She marks a straight baseline AB of length 120 m across the lake and measures the width of the lake, perpendicular to AB, at intervals of 20 m along it, as shown in the following diagram.

Outline of a lake with a straight baseline AB of length 120 m. Perpendicular widths across the lake are measured at 20 m intervals along AB: 34 m, 52 m, 61 m, 48 m and 30 m. The lake's width is 0 m at A and at B. — Diagram not to scale.

(a)
Write down the value of $h$ , the width of each interval, that Ngozi should use in the trapezoidal rule.
1 mark
(b)
Use the trapezoidal rule to estimate the area of the surface of the lake.
3 marks
(c)
Suggest how Ngozi could obtain a more accurate estimate of the area, still using the trapezoidal rule.
1 mark

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Question 5 · AI-01346Paper 2 · 16 marks

An engineer is designing the entrance to a road tunnel. The cross-section of the tunnel is bounded by the road, two vertical walls and a curved roof, as shown in the diagram.

The road lies along the $x$ -axis and the walls lie along $x = 0$ and $x = 12$ . The roof is modelled by the curve

y = - 0.025 x^{3} + 0.3 x^{2} + 3, 0 \leq x \leq 12,

where $x$ and $y$ are measured in metres.

roof

Cross-section of the tunnel. The road is the x-axis and the walls are x = 0 and x = 12.

(a)
Find $\frac{d y}{d x}$ .
2 marks
(b)
Hence find the maximum height of the tunnel.
4 marks
(c)
The table shows some heights of the roof.
$x$ (m)
0
3
6
9
12
$y$ (m)
3
$p$
8.4
$q$
3
Find the value of $p$ and the value of $q$ .
2 marks
(d)
Use the trapezoidal rule with the values in the table to estimate the area of the cross-section of the tunnel.
3 marks
(e)
(i) Write down an integral for the area of the cross-section of the tunnel.
(ii) Hence find the area of the cross-section.
3 marks
(f)
Find the percentage error in the estimate from part (d).
2 marks

$x$ (m)	0	3	6	9	12
$y$ (m)	3	$p$	8.4	$q$	3

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Question 6 · AI-01771Paper 3 · HL only · 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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