IB Maths AI HL · Unit 5: Calculus
IB Maths AI HL Advanced Integration and Kinematics Questions
Exam-style IB Maths AI HL advanced integration and kinematics questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 17 questions
- Paper 1: 17
- 5 easy
- 5 medium
- 5 hard
- 2 starter
- 3 worked examples
Practise Advanced Integration and Kinematics questions →
AI HL formula booklet
What's examined in AI HL advanced integration and kinematics
The question bank covers these advanced integration and kinematics question types (number of questions in brackets):
- Kinematics Problems (6)
- Geometric Applications (6)
- Integration Methods (5)
Key formulas
- Trapezoidal rule (refresher)
- \(A \approx \tfrac{h}{2}\bigl(y_0 + y_n + 2\sum_{i=1}^{n-1} y_i\bigr)\)
In the same notation as the IB formula booklet. All AI HL formulas →
Advanced Integration and Kinematics worked examples
Worked example 1: Evaluating a Definite Integral for Area · easy
The curve $y = -x^2 + 6x - 5$ forms an enclosed region with the $x$-axis. Given that the curve intersects the $x$-axis at $x = 1$ and $x = 5$, calculate the exact area of this enclosed region.
1. Set up the definite integral for the area under the curve between the given roots: $\text{Area} = \int_{1}^{5} (-x^2 + 6x - 5) \, dx$.
2. Find the anti-derivative: $\left[ -\frac{x^3}{3} + \frac{6x^2}{2} - 5x \right]_{1}^{5} = \left[ -\frac{x^3}{3} + 3x^2 - 5x \right]_{1}^{5}$.
3. Substitute the upper limit ($5$): $-\frac{125}{3} + 3(25) - 25 = -\frac{125}{3} + 50 = \frac{25}{3}$.
4. Substitute the lower limit ($1$): $-\frac{1}{3} + 3(1) - 5 = -\frac{1}{3} - 2 = -\frac{7}{3}$.
5. Subtract the evaluations to find the exact area: $\frac{25}{3} - \left(-\frac{7}{3}\right) = \frac{32}{3}$. Area is $\frac{32}{3}$ square units.
Examiner tip: If this appears on Paper 2 or Paper 3, you should use the numerical integration tool on your Graphic Display Calculator instead of integrating by hand to save valuable time.
Worked example 2: Finding Total Distance in Kinematics · medium
The velocity of a particle, $v$ in $\text{m s}^{-1}$, is given by $v(t) = t^2 - 7t + 10$ for $t \ge 0$. Find the exact total distance travelled by the particle during the first $6$ seconds.
1. Identify when the particle is momentarily at rest by setting $v(t) = 0$: $t^2 - 7t + 10 = 0 \implies (t-2)(t-5) = 0$. The particle rests at $t=2$ and $t=5$.
2. Recognize that the velocity is negative between $t=2$ and $t=5$, meaning the particle moves backwards. Total distance requires the absolute value of velocity.
3. Set up the total distance integral, splitting at the sign changes: $D = \int_{0}^{2} v \, dt \;-\; \int_{2}^{5} v \, dt \;+\; \int_{5}^{6} v \, dt$.
4. Evaluate using the antiderivative $F(t) = \frac{t^3}{3} - \frac{7t^2}{2} + 10t$: the three segments give $\frac{26}{3}$, $\frac{9}{2}$, and $\frac{11}{6}$ respectively.
5. Sum the three positive contributions: $\frac{26}{3} + \frac{9}{2} + \frac{11}{6} = \frac{52 + 27 + 11}{6} = \frac{90}{6} = 15$.
6. The total distance travelled is exactly $15\text{ m}$.
Examiner tip: Displacement is $\int v(t) \, dt$, whereas Total Distance is $\int |v(t)| \, dt$. Forgetting the absolute value signs will wrongly subtract the backwards movement from the total.
Worked example 3: Volume of Revolution between Two Curves · hard
Find the exact volume of the solid formed when the region completely enclosed between the curves $y = \sqrt{x}$ and $y = x^2$ is rotated $360^\circ$ around the $x$-axis.
1. Find the points of intersection to establish the limits: $\sqrt{x} = x^2 \implies x = x^4 \implies x(x^3 - 1) = 0$. The limits are $x=0$ and $x=1$.
2. Identify the upper and lower curves on the interval $[0, 1]$: $y = \sqrt{x}$ is above $y = x^2$.
3. Set up the volume of revolution integral using $V = \pi \int (y_{upper}^2 - y_{lower}^2) \, dx$: $V = \pi \int_{0}^{1} \left( (\sqrt{x})^2 - (x^2)^2 \right) \, dx$.
4. Simplify the integrand algebraically: $V = \pi \int_{0}^{1} (x - x^4) \, dx$.
5. Integrate the polynomial: $V = \pi \left[ \frac{x^2}{2} - \frac{x^5}{5} \right]_{0}^{1}$.
6. Evaluate the exact volume: $\pi \left( \frac{1}{2} - \frac{1}{5} \right) = \pi \left( \frac{5}{10} - \frac{2}{10} \right) =$ $\frac{3\pi}{10}$.
Examiner tip: When finding the volume between two curves, you must square the functions individually before subtracting them ($\int (y_1^2 - y_2^2)$). A critical error is subtracting first and then squaring ($\int (y_1 - y_2)^2$).
Try these IB Maths AI HL advanced integration and kinematics questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 4 marks · Paper 1
Find the indefinite integral:
\[\int (4x^3 - 6x^2 + 5) \, dx\]
Question 2 · medium · 5 marks · Paper 1
Find the exact area of the region totally bounded by the curve \(y = 4x - x^2\) and the \(x\)-axis.
Question 3 · hard · 8 marks · Paper 1
A particle moves in a straight line with an acceleration given by \(a(t) = 3t^2 - 14t + 8 \text{ ms}^{-2}\) for \(t \ge 0\).
Given that the initial velocity of the particle is \(3 \text{ ms}^{-1}\), find an expression for its velocity \(v(t)\), and determine the exact interval of time during which the velocity of the particle is strictly decreasing.
All 17 advanced integration and kinematics questions with mark schemes →
FAQ
How many IB Maths AI HL advanced integration and kinematics questions are there?
There are 17 exam-style advanced integration and kinematics questions in the AI HL question bank (Paper 1: 17), graded 5 easy, 5 medium, 5 hard, 2 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is advanced integration and kinematics on Paper 1 or Paper 2?
In the question bank these questions are set as Paper 1 questions.
Where can I get the mark schemes?
Open the AI HL Unit 5 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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