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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

Learn9FM0Option · Further Mechanics 2Further centres of massCentres of mass by calculus

Further centres of mass · Subtopic 1 of 1

Centres of Mass by Integration

Slice the body into thin strips or discs, treat each as a particle, and add with integrals: for a lamina under $y=f(x)$ , $\bar x=\frac{\int xy\,dx}{\int y\,dx}$ ; for a solid of revolution, $\bar x=\frac{\int xy^2\,dx}{\int y^2\,dx}$ . Standard results: cone $\frac h4$ from the base, hemisphere $\frac{3r}8$ from the flat face.

Edexcel 9FM0 Option≈ 50 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Find centres of mass of laminas by integration.
Find centres of mass of solids of revolution.
Use standard solids in composite bodies and toppling problems.

Pause and think

Why is a cone's centre of mass nearer its base than the triangle's centroid ( $\frac13$ height)?

Section 2 of 14

Core idea

Start hereThe toy that won’t fall over

A roly-poly toy always rocks back upright. Its secret is a heavy hemispherical base that keeps the centre of mass low. To find that point exactly, we cut the solid into thin discs and add them up with integration.

The big idea

Treat each thin slice as a particle at its own centre and replace sums by integrals: lamina $\bar x=\frac{\int xy\,dx}{\int y\,dx}$ ; solid of revolution $\bar x=\frac{\int xy^2\,dx}{\int y^2\,dx}$ .

1. The formulas

Solid of revolution

$\delta m=\rho\pi y^2\,\delta x$
Mass of a thin disc at position $x$ .
$\bar x\sum\delta m=\sum x\,\delta m$
Moments, as for particles.
$\bar x=\frac{\int\rho\pi xy^2\,dx}{\int\rho\pi y^2\,dx}$
Let $\delta x\to0$ .
$=\frac{\int xy^2\,dx}{\int y^2\,dx}$
$\rho\pi$ cancels.

Worked exampleSolid cone

Rotate $y=\frac rhx$ , $0\le x\le h$ , about the $x$ -axis. Find $\bar x$ .

Goal: Distance from the vertex.
Strategy: Use the solid formula; constants cancel.

$\int_0^hx\cdot x^2\,dx=\frac{h^4}4$
Numerator (drop $\frac{r^2}{h^2}$ ).
$\int_0^hx^2\,dx=\frac{h^3}3$
Denominator.
$\bar x=\frac34h$
Divide.

Answer

\frac34h

from the vertex, i.e.

\frac h4

from the base.

✓ CheckNearer the wide end ✓.

2. Using standard results

\text{hemisphere: }\tfrac38r;\quad\text{cone: }\tfrac14h;\quad\text{(both from the flat face)}

$r,h$: radius and height.

Worked exampleToppling cylinder

A cylinder, radius $3$ cm, height $10$ cm, stands on a rough plane tilted slowly. Assuming no sliding, when does it topple?

Goal: Angle.
Strategy: Topples when $G$ is above the lowest edge.

$G\ \text{is }5\text{ cm above the base}$
Symmetry.
$\tan\alpha=\frac35$
Vertical through $G$ meets the base edge.

Answer

\alpha\approx31.0^\circ

✓ CheckA squatter cylinder (larger radius) would need a bigger angle ✓.

Lamina or solid?

Drawing diagram…

Choose the formula by the shape of the slice.

Common misconceptionLamina ȳ For a lamina

\bar y=\frac{\int\frac12y^2\,dx}{\int y\,dx}

: each strip's centre is half way up.

Edge caseShells vs solids A hollow hemispherical shell has

G

\frac r2

, not

\frac{3r}8

Pause and think

Why does $\pi$ never appear in the final answer?

Summary

Lamina: weight $y$ ; solid: weight $y^2$ .
Cone $\frac h4$ , hemisphere $\frac{3r}8$ from the base.
Toppling: $G$ over the edge.

Solid of revolution: Solid formed by rotating a region about an axis.
Topple: Tip over about an edge.

HistoryPappus. Pappus of Alexandria (c. 300 AD) linked volumes of revolution to centres of mass: volume = area × distance travelled by the centroid.

Fun factWobbly toys. Roly-poly toys are a hemisphere with a light cone or cylinder on top: the centre of mass sits low, so they always rock back upright.

Section 3 of 14

Key formulae

Lamina

\bar x=\frac{\int xy\,dx}{\int y\,dx},\quad\bar y=\frac{\int\frac12y^2\,dx}{\int y\,dx}

Solid of revolution

\bar x=\frac{\int xy^2\,dx}{\int y^2\,dx}

Standard solids

\text{cone: }\tfrac h4\text{ from base};\quad\text{hemisphere: }\tfrac{3r}8\text{ from face}

TrickTopple or slide? On a rough slope a body topples first if the vertical through

G

passes outside the base before friction gives way: compare

\tan\alpha

with

\frac{\text{half-width}}{\text{height of }G}

and with

\mu

Section 4 of 14

Worked examples

Example 1 · Lamina0 / 3 steps

Find

\bar x

for the region under

y=x^2

0\le x\le2

Example 2 · Cone0 / 2 steps

Show the cone from rotating

y=\frac rhx

(

0\le x\le h

) has

\bar x=\frac34h

from the vertex.

Section 5 of 14

Explore the graph

Centre of the region under y = x^n

Loading the interactive graph…

Try this

The line marks $\bar x=\frac{(n+1)b}{n+2}$ .
$n=0$ (a rectangle) gives $\frac b2$ ; larger $n$ pushes mass right.

$\bar x=\frac{\int x\cdot x^n}{\int x^n}$ .

In the real worldShip stability. Naval architects integrate over hull cross-sections to find the centre of mass and centre of buoyancy, which decide whether a ship rights itself.

Section 6 of 14

Common mistakes

IncorrectUsing

y

instead of

y^2

for solids

CorrectVolume element

\pi y^2\,dx

Discs, not strips.

Incorrect

\bar y=\frac{\int y^2\,dx}{\int y\,dx}

Correct

\frac{\int\frac12y^2\,dx}{\int y\,dx}

Each strip's centre is at half height.

IncorrectCone

\frac h4

from the vertex

CorrectFrom the base

\frac34h

from the vertex.

Hemisphere centre of mass: r/2?

3r/8. Integrate, don't guess.

Section 7 of 14

Exam tip

Exam tipπ cancels. In solids of revolution

\pi

appears top and bottom; leave it out from the start.

Section 8 of 14

Guided practice

SA-FM-CS-G013 marksL2 StandardNon-calculatorLaminas by integration

Find

\bar x

for the uniform lamina under

y=x^2

0\le x\le2

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FM-CS-0012 marksL2 StandardNon-calculatorSolids

A solid cone has height

12

cm. How far is its centre of mass from its base?

Salam Academy Original Practice · Proposed mark scheme

SA-FM-CS-0021 markL2 StandardNon-calculatorSolids

A solid hemisphere has radius

8

cm. How far is its centre of mass from the flat face?

Salam Academy Original Practice · Proposed mark scheme

SA-FM-CS-0033 marksL3 ExamNon-calculatorSolids

The region under

y=\sqrt x

0\le x\le4

is rotated about the

x

-axis. Find

\bar x

for the solid.

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-FM-CS-0043 marksL4 ChallengingCalculatorToppling

A uniform solid cylinder of radius

3

cm and height

10

cm stands on a rough plane. The plane is slowly tilted; friction is enough to stop sliding. Find the angle (degrees) at which it topples.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Lamina: $\frac{\int xy}{\int y}$ .
Solid: $\frac{\int xy^2}{\int y^2}$ .
Cone $\frac h4$ , hemisphere $\frac{3r}8$ from the base.

Section 12 of 14

Quick quiz

Q1 · SA-FM-CS-Q011 markL1 FoundationNon-calculatorSolids

For a solid of revolution about the

x

-axis,

\bar x

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: A level Further Mathematics 9FM0
Where: Further Mechanics 2 · Further centres of mass
Expected: Centres of mass of uniform laminas and solids of revolution by integration; standard solids (hemisphere, cone); composite solids; toppling on inclined planes.
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 · Option

Section 14 of 14

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