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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9FM0Paper 1 · Core Pure Mathematics 1Further calculusVolumes, mean values and improper integrals

Further calculus · Subtopic 1 of 3

Volumes, Mean Values and Improper Integrals

Spin a curve about an axis and the solid's volume is $\pi\int y^2\,dx$ (or $\pi\int x^2\,dy$ ). The mean value of $f$ on $[a,b]$ is $\frac1{b-a}\int_a^bf\,dx$ . An improper integral has an infinite limit or an infinite integrand: replace it by a limit and see whether that limit exists.

Edexcel 9FM0 Paper 1Edexcel 9FM0 Paper 2AQA 7367 Paper 1AQA 7367 Paper 2≈ 45 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Find volumes of revolution about the $x$ - and $y$ -axes.
Find the mean value of a function over an interval.
Evaluate improper integrals as limits and decide whether they converge.

Pause and think

Can an infinitely long region have a finite area?

Section 2 of 14

Core idea

Start herePaint that fills but can't cover

Spin $y=\frac1x$ (for $x\ge1$ ) around the $x$ -axis and you get Gabriel's horn: infinitely long, yet its volume is exactly $\pi$ . Integrals over infinite ranges can still have finite values, if the integrand fades fast enough.

The big idea

$V_x=\pi\int y^2\,dx$ , $V_y=\pi\int x^2\,dy$ . Mean value $\frac1{b-a}\int_a^bf\,dx$ . Improper integrals are limits: $\int_a^\infty f=\lim_{b\to\infty}\int_a^bf$ .

1. Volumes of revolution

V=\pi\int y^2\,dx

$\delta V\approx\pi y^2\,\delta x$
A thin slice is a disc of radius $y$ , thickness $\delta x$ .
$V=\lim\sum\pi y^2\,\delta x=\pi\int_a^by^2\,dx$
Add the discs.

About the

y

-axis use discs of radius

x

\pi\int x^2\,dy

Worked example $y=x^2$ about the $y$ -axis, $0\le y\le2$

Volume.

Goal: $V$ .
Strategy: Express $x^2$ in terms of $y$ .

$x^2=y$
From the curve.
$V=\pi\int_0^2y\,dy=\pi\left[\tfrac{y^2}2\right]_0^2$
Discs.
$=2\pi$
Evaluate.

Answer

2\pi

✓ CheckHalf the cylinder of radius

\sqrt2

, height

2

(

4\pi

): a paraboloid fills exactly half ✓.

2. Mean value

\bar f=\frac1{b-a}\int_a^bf(x)\,dx

$\bar f$: the height of the rectangle with the same area and width.

3. Improper integrals

\int_1^\infty x^{-p}\,dx

$\int_1^bx^{-p}dx=\frac{b^{1-p}-1}{1-p}$
For $p\neq1$ .
$p>1:\ b^{1-p}\to0\Rightarrow\frac1{p-1}$
Converges.
$p<1:\ b^{1-p}\to\infty$
Diverges.
$p=1:\ \ln b\to\infty$
Diverges.

Worked example $\int_0^1\frac{dx}{\sqrt x}$

Evaluate.

Goal: Value.
Strategy: Integrand infinite at $0$ : limit from the right.

$\lim_{a\to0^+}\int_a^1x^{-1/2}dx$
Write as a limit.
$=\lim_{a\to0^+}\left(2-2\sqrt a\right)=2$
$\sqrt a\to0$ .

Answer

2

✓ CheckHere

p=\frac12<1

, so the integral near

0

converges ✓.

Common misconception"Plugging in $\infty$ " Write the limit explicitly and say what happens to the term in

b

Summary

Disc method about either axis.
Mean value: area ÷ width.
Improper integrals as limits.

Converges: The limit exists and is finite.
Diverges: No finite limit.

HistoryGabriel's horn. Evangelista Torricelli (1643) showed that rotating

y=\frac1x

for

x\ge1

gives a solid of finite volume

\pi

but infinite surface area: you could fill it with paint but not paint it.

Fun factAverage temperature. Weather services report the mean value of the temperature function over a day: literally

\frac1{24}\int_0^{24}T(t)\,dt

Section 3 of 14

Key formulae

Volumes

V_x=\pi\int_a^by^2\,dx,\qquad V_y=\pi\int_c^dx^2\,dy

Mean value

\bar f=\frac1{b-a}\int_a^bf(x)\,dx

Improper integral

\int_a^\infty f(x)\,dx=\lim_{b\to\infty}\int_a^bf(x)\,dx

TrickPowers of 1/x.

\int_1^\infty\frac{dx}{x^p}

converges exactly when

p>1

;

\int_0^1\frac{dx}{x^p}

converges exactly when

p<1

Section 4 of 14

Worked examples

Example 1 · Improper integral0 / 2 steps

Evaluate

\int_1^\infty\frac{dx}{x^2}

Example 2 · About the y-axis0 / 2 steps

y=x^2

0\le y\le2

, rotated about the

y

-axis. Find the volume.

Section 5 of 14

Explore the graph

Area under 1/x² out to b

Loading the interactive graph…

Try this

The horizontal line shows the area $1-\frac1b$ from $1$ to $b$ .
As $b$ grows, it creeps up to $1$ but never passes it.

$\int_1^b\frac{dx}{x^2}=1-\frac1b\to1$ .

In the real worldEscape velocity. The work to escape a planet is

\int_R^\infty\frac{GMm}{r^2}\,dr=\frac{GMm}R

: an improper integral with a finite answer.

Section 6 of 14

Common mistakes

Incorrect

V=\pi\int y\,dx

Correct

\pi\int y^2\,dx

Each slice is a disc of area

\pi y^2

IncorrectSubstituting

\infty

directly

CorrectUse a limit

\infty

is not a number; show the limit.

Incorrect

\int_1^\infty\frac{dx}x

converges

CorrectIt diverges

\ln b\to\infty

Infinitely long region

Area: 1. Exactly 1.

Section 7 of 14

Exam tip

Exam tipWrite the limit. Mark schemes give a mark for writing

\lim_{b\to\infty}

and showing what happens to the term in

b

Section 8 of 14

Guided practice

SA-FP-VI-G012 marksL1 FoundationCalculatorVolumes

The line

y=x

for

0\le x\le3

is rotated

2\pi

about the

x

-axis. Find the volume (to 3 s.f.).

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FP-VI-0012 marksL2 StandardNon-calculatorImproper integrals

Evaluate

\displaystyle\int_1^\infty\frac{dx}{x^2}

Salam Academy Original Practice · Proposed mark scheme

SA-FP-VI-0022 marksL2 StandardNon-calculatorImproper integrals

Evaluate

\displaystyle\int_0^1\frac{dx}{\sqrt x}

Salam Academy Original Practice · Proposed mark scheme

SA-FP-VI-0032 marksL2 StandardNon-calculatorMean value

Find the mean value of

x^2

over

0\le x\le3

Salam Academy Original Practice · Proposed mark scheme

SA-FP-VI-0042 marksL2 StandardCalculatorVolumes

y=\sqrt x

0\le x\le4

, is rotated about the

x

-axis. Find the volume (to 3 s.f.).

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-FP-VI-0053 marksL3 ExamCalculatorVolumes

The curve

y=x^2

0\le y\le2

, is rotated about the

y

-axis. Find the volume (to 3 s.f.).

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$\pi\int y^2dx$ or $\pi\int x^2dy$ .
Mean value $\frac1{b-a}\int f$ .
Improper: limits; check convergence.

Section 12 of 14

Quick quiz

Q1 · SA-FP-VI-Q011 markL1 FoundationNon-calculatorImproper integrals

Which integral diverges?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: A level Further Mathematics 9FM0
Where: Core Pure · Further calculus
Expected: Volumes of revolution about x- and y-axes (including parametric), mean value, improper integrals with infinite limits or undefined integrands.
Calculator: Permitted; show enough working to earn the method marks.
Typical style: Edexcel often opens with a routine part and builds to a "show that" or "prove" part.
Open in: 9FM0 · Paper 1 9FM0 · Paper 2

Section 14 of 14

Past paper questions

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel 9FM0, First teaching 2017 (official specification).