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 $a$ .

Rewrite $\frac{1}{x^2}$ as $x^{-2}$ . $\int_1^a x^{-2} \,dx = \left[ -x^{-1} \right]_1^a = \left[ -\frac{1}{x} \right]_1^a$ . (M1) $= \left(-\frac{1}{a}\right) - \left(-\frac{1}{1}\right) = 1 - \frac{1}{a}$ . A1 Set the expression equal to $0.8$ : $1 - \frac{1}{a} = 0.8$ . (M1) $\frac{1}{a} = 1 - 0.8 \implies \frac{1}{a} = 0.2$ . (M1) $a = \frac{1}{0.2} = 5$ . A1

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IB Maths AA SL · Unit 5: Calculus

Definite Integrals & Area (IB Maths AA SL): notes, exam questions and worked examples

Updated October 2026 · By Pete Bromfield

A definite integral $\int_{a}^{b} f (x) d x = F (b) - F (a)$ gives a number, not a function, so there is no $+ C$ . It measures the signed area between the curve and the $x$ -axis: area below the axis counts as negative, so split the integral at the roots or use $\int_{a}^{b} | y | d x$ . The area between two curves is $\int_{a}^{b} | f (x) - g (x) | d x$ between their intersections.

32 questions
Paper 1: 23
Paper 2: 9
3 worked examples

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Definite Integrals & Area notes

Evaluating a definite integral

Integrate, then substitute the upper limit and subtract the value at the lower limit: $\int_{1}^{3} 2 x d x = [x^{2}]_{1}^{3} = 9 - 1 = 8$ . Use brackets when you substitute; most slips here are sign errors.

Area between a curve and the x-axis

Where the curve is below the axis the integral is negative. For $y = x^{2} - 1$ from 0 to 2, $\int_{0}^{2} (x^{2} - 1) d x = \frac{2}{3}$ , but the area is $\frac{2}{3} + \frac{4}{3} = 2$ : find the root at $x = 1$ and add the two areas as positive values. Sketch the curve first.

Area between two curves

Find where the curves meet, then integrate the top curve minus the bottom one between those $x$ -values. Between $y = x$ and $y = x^{2}$ , the curves meet at $x = 0$ and $x = 1$ , and the area is $\int_{0}^{1} (x - x^{2}) d x = \frac{1}{6}$ .

Unknown limits

Set the integral equal to the given value and solve. $\int_{0}^{k} 2 x d x = 9$ gives $k^{2} = 9$ , so $k = 3$ when $k > 0$ .

Definite integrals on the GDC (Paper 2)

On Paper 2 the GDC evaluates a definite integral directly. Write down the integral you entered, with its limits, then the value: the integral earns the method mark even if the number is wrong.

Key formulas

Integral of $x^{n}$ , $n \neq - 1$ In the booklet: $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C$
Definite integral Learn it: $\int_{a}^{b} f^{'} (x) d x = f (b) - f (a)$
Area, curve and $x$ -axis In the booklet: $A = \int_{a}^{b} | y | d x$
Area between two curves Learn it: $A = \int_{a}^{b} | f (x) - g (x) | d x$

The IB formula booklet is provided in every exam paper. In the booklet = printed there, so know where to find it; Learn it = not in the booklet, so learn it by heart. All AA SL formulas →

Definite Integrals & Area worked examples

Worked example 1: Finding an unknown upper limit · easy

Find the exact value of the positive constant $k$ such that $\int_{0}^{k} 3 x^{2} d x = 64$ .

Solution

6 steps · 1 to fill in

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.

Solution

6 steps · 1 to fill in

Examiner tip: The area between two curves is always

\int_{a}^{b} (Top - Bottom) d x

. 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 $y = x^{2} - 3 x + 4$ and $y = 4 - x^{2} + 2 x$ . 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.

Solution

6 steps · 1 to fill in

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.

Definite Integrals & Area exam-style questions

3 questions from our AA SL bank, written in the style of IB papers, easiest first. One has its full worked solution open here. Open the worked answer, or check your answer 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 $\int_{0}^{4} (\frac{1}{2} x^{2} + 2) d x$ .
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.

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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 $\int_{1}^{a} \frac{1}{x^{2}} d x$ in terms of $a$ .
Set up an equation and solve it analytically to find the exact value of $a$ .

Full worked solution Free, no sign-in

Rewrite $\frac{1}{x^{2}}$ as $x^{- 2}$ .
$\int_{1}^{a} x^{- 2} d x = {[- x^{- 1}]}_{1}^{a} = {[- \frac{1}{x}]}_{1}^{a}$ . (M1)
$= (- \frac{1}{a}) - (- \frac{1}{1}) = 1 - \frac{1}{a}$ . A1
Set the expression equal to $0.8$ : $1 - \frac{1}{a} = 0.8$ . (M1)
$\frac{1}{a} = 1 - 0.8 ⟹ \frac{1}{a} = 0.2$ . (M1)
$a = \frac{1}{0.2} = 5$ . A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 5 marks · Paper 1

Find the exact value of the following definite integral, giving your answer in the form $p \ln q$ : $\int_{1}^{5} \frac{3}{2 x} d x$

Attempt it and see the mark scheme →

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Where students lose marks on definite integrals & area

Examiner's noteIntegration: missing +C, substituting into the derivative, and area limits

Where marks are lost: Given a derivative and a point, candidates substituted the point into the derivative, differentiated instead of integrating, or dropped the constant. Integrating e^(kx) without the 1/k, not recognising ∫ 1/x dx = ln x, not writing the final function, and substituting only one limit in an area were all reported.

Examiners flagged this in 4 exam series (May 2024, November 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Definite Integrals & Area on your calculator

A GDC is allowed on Paper 2, not on Paper 1. Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire, with a checked worked example:

Numerical derivative and integral on your calculator

Definite Integrals & Area FAQ

Is the area formula in the IB formula booklet?

The AA SL booklet gives the area between a curve and the x-axis, A = ∫|y| dx. The area between two curves is not printed: integrate the top curve minus the bottom curve between the points where they meet.

Why is my area negative?

Part of the curve is below the x-axis, where the integral is negative. Split the integral at the roots, or integrate |y|, and add the areas as positive values.

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, 17 medium, 7 hard, 3 very hard. 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.

Next steps

More AA SL Unit 5: Applications of Differentiation · Graphical Behaviour and the Second Derivative · Integration · Limits and Rates of Change · Differentiation Rules

Teaching this topic? Free ready-to-teach lesson (slides and teacher notes): Definite integrals & area under a curve.

Keep revising AA SL: AA SL revision notes · IB Maths AA SL past papers · AA SL common exam mistakes · AA SL syllabus checklist · AA SL calculus knowledge organiser

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Definite Integrals & Area (IB Maths AA SL): notes, exam questions and worked examples

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Definite Integrals & Area notes

Evaluating a definite integral

Area between a curve and the x-axis

Area between two curves

Unknown limits

Definite integrals on the GDC (Paper 2)

Key formulas

Definite Integrals & Area worked examples

Worked example 1: Finding an unknown upper limit · easy

Solution

Worked example 2: Area between a line and a curve · medium

Solution

Worked example 3: Area between two intersecting parabolas · hard

Solution

Definite Integrals & Area exam-style questions

Question 1 · easy · 4 marks · Paper 2

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 5 marks · Paper 1

Where students lose marks on definite integrals & area

Definite Integrals & Area on your calculator

Definite Integrals & Area FAQ

Next steps

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