5.5 Introduction to integration questions for IB Maths AI

51 questions on Introduction to integration, 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	16	0	16
Paper 2	27	8	35
Every paper	43	8	51

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-01186Paper 1 · 6 marks

Consider the function $f$ whose gradient function is $f^{'} (x) = 6 x^{2} - 4 x + 5$ .

(a)
Find $\int (6 x^{2} - 4 x + 5) d x$ .
3 marks
(b)
The graph of $f$ passes through the point $(1, 10)$ . Find $f (x)$ .
3 marks

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Question 2 · AI-00251Paper 2 · HL only · 15 marks

A train leaves a station along a straight track. Its velocity, $v$ m s⁻¹, $t$ seconds after leaving the station, is modelled by

v (t) = {\begin{cases} 0.03 t^{2}, & 0 \leq t \leq 20 \\ p t + q, & 20 < t \leq 50 \\ 48, & t > 50 \end{cases}

where $p$ and $q$ are constants. Both $v$ and its gradient $\frac{d v}{d t}$ are continuous at $t = 20$ .

(a)
Find the value of $p$ and the value of $q$ .
4 marks
(b)
Show that the velocity of the train is continuous at $t = 50$ .
1 mark
(c)
Write down the acceleration of the train when $t = 35$ .
2 marks
(d)
Find the distance travelled by the train in the first 50 seconds.
4 marks
(e)
The next station is 5 km from the first. Assuming the train does not slow down, find the time it takes the train to travel the 5 km.
3 marks
(f)
Comment on whether the model is realistic at $t = 50$ .
1 mark

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Question 3 · AI-01187Paper 1 · 6 marks

Find each of the following integrals.

(a)
$\int (2 x + 3) (x - 1) d x$
3 marks
(b)
$\int \frac{x^{3} - 4}{x^{2}} d x$
3 marks

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Question 4 · AI-00424Paper 2 · HL only · 18 marks

A robotic floor cleaner is programmed to follow a spiral path. Its position vector, in metres, relative to its charging dock, $t$ seconds after it starts is

r = (\begin{array}{c} 0.1 t \cos t \\ 0.1 t \sin t \end{array}), 0 \leq t \leq 4 π .

(a)
Find the velocity vector of the cleaner at time $t$ .
3 marks
(b)
Show that the speed of the cleaner is $0.1 \sqrt{1 + t^{2}}$ .
3 marks
(c)
Find the speed of the cleaner when $t = 2 π$ .
2 marks
(d)
Find the first time $t > 0$ at which the cleaner is moving parallel to the $x$ -axis, and its distance from the charging dock at that time.
4 marks
(e)
Find the magnitude of the acceleration of the cleaner when $t = π$ .
4 marks
(f)
Find the distance travelled by the cleaner in the first $2 π$ seconds.
2 marks

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Question 5 · AI-01188Paper 1 · 4 marks

The following diagram shows part of the graph of $y = - x^{2} + 6 x + 1$ . The region R is enclosed by the curve, the $x$ -axis and the lines $x = 1$ and $x = 5$ .

y = −x² + 6x + 1

Region R lies under the curve between x = 1 and x = 5.

(a)
Write down an integral that represents the area of R.
2 marks
(b)
Find the area of R.
2 marks

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Question 6 · AI-00426Paper 2 · HL only · 16 marks

A survey drone flies at a constant height above a field. Its horizontal velocity, in m s $^{- 1}$ , $t$ seconds after it starts is

v = (\begin{array}{c} 3 \sin (0.5 t) \\ 4 - 0.5 t \end{array}), 0 \leq t \leq 10,

where the first component is east and the second is north. The drone starts directly above the origin O.

(a)
Find the speed of the drone when $t = 3$ .
2 marks
(b)
Find the value of $t > 0$ at which the drone is moving due north.
3 marks
(c)
Find the acceleration of the drone when $t = 2$ .
2 marks
(d)
Find an expression for the position vector of the drone relative to O at time $t$ .
4 marks
(e)
Find the distance of the drone from its starting point when $t = 10$ .
2 marks
(f)
Find the total distance flown by the drone during the 10 seconds.
2 marks
(g)
Explain why your answer to part (f) is greater than your answer to part (e).
1 mark

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