Find the exact value of $\int_{4}^{9} \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right) \, dx$.

M1 for writing as fractional indices: $\int (x^{\frac{1}{2}} + x^{-\frac{1}{2}}) \, dx$. M1 for integrating to get $[\frac{2}{3}x^{\frac{3}{2}} + 2x^{\frac{1}{2}}]$. M1 for substituting the limits: $(\frac{2}{3}(27) + 2(3)) - (\frac{2}{3}(8) + 2(2))$. A1 for evaluating to get exactly $14\frac{2}{3}$ (or $\frac{44}{3}$).

A Level Math Revision

Edexcel IAL P2 Definite Integration and Area: exam questions and answers, notes and worked examples

Q: Is the trapezium rule in the IAL formula booklet?

Yes: on our IAL formula sheet the trapezium rule is marked In the formula booklet. The area under a curve as a definite integral is marked Learn it.

Q: How do I know if the trapezium rule over- or under-estimates?

Look at the curve's shape. If it bends upwards (convex) the straight tops of the trapezia lie above it and the rule overestimates; if it bends downwards it underestimates.

Q: Why does my definite integral give 0 when the area is not 0?

Because the curve crosses the x-axis between the limits, and the parts below the axis count as negative. For y = x³ − 4x from x = −2 to x = 2 the integral is 0, but the curve also crosses the axis at x = 0: the integral from −2 to 0 is 4 and from 0 to 2 is −4, so the area is 4 + 4 = 8. Find the roots first, integrate each part separately and add the sizes.

Q: Is a calculator allowed in IAL P2?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P2 included. Method marks are for the working you write down, so show it.

Q: How long is the IAL P2 exam?

Each IAL maths unit, P2 included, is one written paper of 1 hour 30 minutes worth 75 marks.

Q: How often does definite integration and area come up in P2?

In our mapping of the 4 Pearson P2 papers from June 2024 to June 2025, definite integration and area was tested in 4 of them (6 questions in all). Any topic can be examined in any series.

Updated October 2026 · By Pete Bromfield

P2 integration evaluates definite integrals, $\int_{a}^{b} f (x) d x = F (b) - F (a)$ , and uses them for areas: under a curve, between a curve and a line, and below the $x$ -axis (where the integral is negative). When the integral cannot be found exactly, the trapezium rule estimates it.

P2 · Pure Mathematics 2
Calculator allowed
3 worked examples
33 practice questions
Tested in 4 of the last 4 papers

Practise P2 definite integration and area →

Notes
Formulas
Worked examples
Mistakes
FAQ

P2 Definite Integration and Area notes

Definite integrals

Integrate, then substitute the limits and subtract: $\int_{1}^{3} (3 x^{2} - 2 x) d x = [x^{3} - x^{2}]_{1}^{3} = 18 - 0 = 18$ . No $+ c$ is needed.

Area under a curve

The integral gives the area between the curve and the $x$ -axis when the curve is above the axis. Below the axis the integral is negative, so find the roots first and treat each part separately: for $y = x^{2} - 4 x$ between $x = 0$ and $x = 4$ the integral is $- \frac{32}{3}$ , so the area is $\frac{32}{3}$ .

Area between a curve and a line

Integrate (top minus bottom) between the $x$ -coordinates where they meet. $y = 2 x$ is above $y = x^{2}$ for $0 < x < 2$ , so the area between them is $\int_{0}^{2} (2 x - x^{2}) d x = \frac{4}{3}$ .

The trapezium rule

With $n$ strips of width $h = \frac{b - a}{n}$ : $\int_{a}^{b} y d x \approx \frac{1}{2} h {(y_{0} + y_{n}) + 2 (y_{1} + \dots + y_{n - 1})}$ . For $\int_{0}^{2} \sqrt{1 + x^{2}} d x$ with 4 strips, $h = 0.5$ and the estimate is $2.98$ (to 3 s.f.). The curve bends upwards here, so the trapezia lie above it and the estimate is too big; more strips make it more accurate.

Key formulas

Area under a curve Learn it: $\int_{a}^{b} y d x$
Trapezium rule, $h = \frac{b - a}{n}$ In the formula booklet: $\int_{a}^{b} y d x \approx \frac{1}{2} h {(y_{0} + y_{n}) + 2 (y_{1} + \dots + y_{n - 1})}$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P2 formulas on one page (free PDF) →

P2 Definite Integration and Area worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 7 marks · hard

Find the exact total area of the finite regions bounded by the curves $y = x^{3} + x^{2}$ and $y = 2 x^{2} + 2 x$ .

Mark scheme

M1 for setting $x^{3} + x^{2} = 2 x^{2} + 2 x ⟹ x^{3} - x^{2} - 2 x = 0$ .
M1 for factoring $x (x - 2) (x + 1) = 0$ to find the intersection points $x = - 1, 0, 2$ .
M1 for recognising two distinct regions and setting up integrals: $\int_{- 1}^{0} (x^{3} - x^{2} - 2 x) d x$ and $\int_{0}^{2} (2 x^{2} + 2 x - x^{3} - x^{2}) d x$ .
M1 for integrating the generic polynomial to $\pm [\frac{x^{4}}{4} - \frac{x^{3}}{3} - x^{2}]$ .
M1 for evaluating the area of the first region to $\frac{5}{12}$ .
M1 for evaluating the area of the second region to $\frac{8}{3}$ .
A1 for calculating the combined total area: $\frac{5}{12} + \frac{8}{3} = \frac{37}{12}$ .

7 steps · 1 to fill in

Worked example 2 · 6 marks · hard

The curve with equation $y = x^{3} - 4 x$ intersects the x-axis at $x = - 2$ , $x = 0$ , and $x = 2$ . Find the total exact area enclosed between the curve and the x-axis.

Mark scheme

M1 for recognising that the region between $0$ and $2$ is below the x-axis, so the integrals must be computed separately.
M1 for integrating to find the generic primitive: $[\frac{1}{4} x^{4} - 2 x^{2}]$ .
M1 for evaluating the integral from $- 2$ to $0$ : $(0) - (4 - 8) = 4$ .
M1 for evaluating the integral from $0$ to $2$ : $(4 - 8) - (0) = - 4$ .
M1 for taking the absolute value of the negative area before combining.
A1 for the total correct area: $4 + | - 4 | = 8$ .

6 steps · 1 to fill in

Worked example 3 · 4 marks · medium

Find the exact value of $\int_{4}^{9} (\sqrt{x} + \frac{1}{\sqrt{x}}) d x$ .

Mark scheme

M1 for writing as fractional indices: $\int (x^{\frac{1}{2}} + x^{- \frac{1}{2}}) d x$ .
M1 for integrating to get $[\frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}}]$ .
M1 for substituting the limits: $(\frac{2}{3} (27) + 2 (3)) - (\frac{2}{3} (8) + 2 (2))$ .
A1 for evaluating to get exactly $14 \frac{2}{3}$ (or $\frac{44}{3}$ ).

4 steps

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Common P2 definite integration and area mistakes

Area below the axis. A region below the x-axis gives a negative integral. Integrate each part separately and add the sizes; integrating straight across the root can cancel areas.
Subtracting the wrong way. It is $F (upper) - F (lower)$ , and between two graphs it is (top) − (bottom).
Miscounting the trapezium ordinates. $n$ strips need $n + 1$ $y$ -values; $h$ is the strip width, not the number of strips.
Rounding the ordinates early. Keep at least 4 decimal places in the y-values, then round the final estimate.

Every P2 mistake from the examiner reports, with the paper it came from →

On your calculator: how to find a numerical derivative and definite integral. A calculator is allowed in every IAL unit.

P2 Definite Integration and Area FAQ

Is the trapezium rule in the IAL formula booklet?

Yes: on our IAL formula sheet the trapezium rule is marked In the formula booklet. The area under a curve as a definite integral is marked Learn it.

How do I know if the trapezium rule over- or under-estimates?

Look at the curve's shape. If it bends upwards (convex) the straight tops of the trapezia lie above it and the rule overestimates; if it bends downwards it underestimates.

Why does my definite integral give 0 when the area is not 0?

Because the curve crosses the x-axis between the limits, and the parts below the axis count as negative. For y = x³ − 4x from x = −2 to x = 2 the integral is 0, but the curve also crosses the axis at x = 0: the integral from −2 to 0 is 4 and from 0 to 2 is −4, so the area is 4 + 4 = 8. Find the roots first, integrate each part separately and add the sizes.

Is a calculator allowed in IAL P2?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P2 included. Method marks are for the working you write down, so show it.

How long is the IAL P2 exam?

Each IAL maths unit, P2 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does definite integration and area come up in P2?

In our mapping of the 4 Pearson P2 papers from June 2024 to June 2025, definite integration and area was tested in 4 of them (6 questions in all). Any topic can be examined in any series.

Next steps

More P2: Sequences and Series · Logarithms and Exponentials · Circles · Further Trigonometry · Algebraic Methods · Proof · The Binomial Expansion · Differentiation (stationary points)

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Edexcel IAL P2 Definite Integration and Area: exam questions and answers, notes and worked examples

Try it first

P2 Definite Integration and Area notes

Definite integrals

Area under a curve

Area between a curve and a line

The trapezium rule

Key formulas

P2 Definite Integration and Area worked examples

Worked example 1 · 7 marks · hard

Mark scheme

Worked example 2 · 6 marks · hard

Mark scheme

Worked example 3 · 4 marks · medium

Mark scheme

Common P2 definite integration and area mistakes

P2 Definite Integration and Area FAQ

Is the trapezium rule in the IAL formula booklet?

How do I know if the trapezium rule over- or under-estimates?

Why does my definite integral give 0 when the area is not 0?

Is a calculator allowed in IAL P2?

How long is the IAL P2 exam?

How often does definite integration and area come up in P2?

Next steps

Your P2 revision kit