IB Maths AA SL · Unit 5: Calculus
IB Maths AA SL Definite Integrals Questions
Exam-style IB Maths AA SL definite integrals 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.
- 32 questions
- Paper 1: 23
- Paper 2: 9
- 5 easy
- 14 medium
- 7 hard
- 3 very hard
- 3 starter
- 3 worked examples
Practise Definite Integrals questions →
AA SL formula booklet
What's examined in AA SL definite integrals
The question bank covers these definite integrals question types (number of questions in brackets):
- Evaluating Definite Integrals (17)
- Area of Bounded Regions (14)
- Properties of Integrals (1)
Key formulas
- Integral of x^n
- \(\int x^n\, dx = \dfrac{x^{n+1}}{n+1} + C,\ n \ne -1\)
- Standard integrals
- \(\int \sin x\, dx = -\cos x + C,\ \int \cos x\, dx = \sin x + C,\ \int e^x\, dx = e^x + C,\ \int \dfrac{1}{x}\, dx = \ln|x| + C\)
- Area between a curve and the x-axis
- \(A = \int_a^b y\, dx\)
- Volume of revolution (about x-axis)
- \(V = \pi \int_a^b y^2\, dx\)
In the same notation as the IB formula booklet. All AA SL formulas →
Definite Integrals worked examples
Worked example 1: Finding an unknown upper limit · easy
Find the exact value of the positive constant $k$ such that $\int_0^k 3x^2 \,dx = 64$.
1. Anti-differentiate the integrand: $\int 3x^2 \,dx = x^3$.
2. Set up evaluation brackets: $\left[ x^3 \right]_0^k$.
3. Substitute the limits: $(k^3) - (0^3) = k^3$.
4. Equate to the given area: $k^3 = 64$.
5. Solve by taking the cube root of both sides.
6. State: $\mathbf{k = 4}$.
Examiner tip: When the unknown is in the limits of integration, evaluate the integral as usual — leave the parameter in the expression — then solve the resulting algebraic equation at the end.
Worked example 2: Area between a line and a curve · medium
Consider the line $y = x + 2$ and the curve $y = x^2$. Use your GDC to find the exact area of the region completely enclosed by the two graphs.
1. Graph both functions on the GDC to identify the enclosed region and the upper boundary.
2. Determine the intersection points (CG50: G-Solv $\to$ ISCT): $x = -1$ and $x = 2$.
3. Identify the line $y = x + 2$ is above the parabola $y = x^2$ on this interval.
4. Set up: $\text{Area} = \int_{-1}^{2} \big((x + 2) - x^2\big) \,dx$.
5. Evaluate using the numerical integration tool on the GDC.
6. State: $\mathbf{\text{Area} = \frac{9}{2} = 4.5}$.
Examiner tip: The area between two curves is always $\int_a^b (\text{Top} - \text{Bottom}) \,dx$. If you subtract in the wrong order the answer is negative — take the absolute value.
Worked example 3: Area between two intersecting parabolas · hard
The curves have equations $y = x^2 - 3x + 4$ and $y = 4 - x^2 + 2x$. Find the $x$-coordinates of the two points where the graphs intersect, and hence calculate the exact area of the region bounded between the curves using your GDC.
1. Set the equations equal: $x^2 - 3x + 4 = 4 - x^2 + 2x$.
2. Rearrange: $2x^2 - 5x = 0$.
3. Factorise: $x(2x - 5) = 0 \implies x = 0$ or $x = 2.5$.
4. Identify the upper curve. $y = 4 - x^2 + 2x$ opens downwards, so it forms the top of the enclosed region.
5. Set up the area integral: $\text{Area} = \int_0^{2.5} \big((4 - x^2 + 2x) - (x^2 - 3x + 4)\big) \,dx = \int_0^{2.5} (5x - 2x^2) \,dx$.
6. Evaluate on the GDC: $\mathbf{\text{Area} = \frac{125}{24} \approx 5.21}$ (3 s.f.).
Examiner tip: In Paper 2, once the intersection limits are algebraically shown you don't need to hand-integrate unless the question explicitly says "Use calculus to...". Evaluate the simplified integral on the GDC.
Try these IB Maths AA SL definite integrals 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 2
Consider the function \(f(x) = \frac{1}{2}x^2 + 2\).
Using your graphic display calculator, calculate the exact value of the definite integral \(\int_0^4 \left(\frac{1}{2}x^2 + 2\right) \,dx\).
A student estimates the area under the curve between \(x = 0\) and \(x = 4\) using a basic geometric method and gets an answer of \(19\). Calculate the percentage error of their approximation.
Question 2 · medium · 5 marks · Paper 1
The area enclosed by the curve \(y = \frac{1}{x^2}\) (for \(x > 0\)), the \(x\)-axis, the vertical line \(x = 1\), and the vertical line \(x = a\) (where \(a > 1\)) is exactly \(0.8\) square units.
Find an expression for \(\int_1^a \frac{1}{x^2} \,dx\) in terms of \(a\).
Set up an equation and solve it analytically to find the exact value of \(a\).
Question 3 · hard · 5 marks · Paper 1
Find the exact value of the following definite integral, giving your answer in the form \(p\ln q\):
\[\int_1^5 \frac{3}{2x} \,dx\]
All 32 definite integrals questions with mark schemes →
FAQ
How many IB Maths AA SL definite integrals questions are there?
There are 32 exam-style definite integrals questions in the AA SL question bank (Paper 1: 23 · Paper 2: 9), graded 5 easy, 14 medium, 7 hard, 3 very hard, 3 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is definite integrals on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 23 · Paper 2: 9. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA SL 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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