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Integration – Volumes of revolution

Department of MathematicsUniversity of Leicester

Content

Around y-axis

Around x-axis

Introduction

Introduction

If a curve is rotated around either the x-axis or y-axis, a solid is formed.

The volume of this solid is called the “Volume of revolution”.

Around y-axisAround x-axis

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Introduction

Examples: click to see the solids formed

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Around y-axis

Around y-axis

Around x-axis

Clear

y

x

𝑓 (𝑥)

Click here to see rotate around the x-axis:

Volume of Revolution around x-axis

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x

y

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x

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y

Introduction

x

y

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Repeat

We approximate the area by rectangular strips.

We write .

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Another way of looking at integration

So is the area of one strip.And is the area of all the strips.

Introduction

Click here to see what each bit

means

We approximate the area by rectangular strips.

We write .

Around y-axisAround x-axis

Another way of looking at integration

So is the area of one strip.And is the area of all the strips.

Introduction

Around y-axisAround x-axis

Another way of looking at integration

Introduction

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∫ means sum over all the strips

  

)(xf

)(xf

dx

dx

dx

 

 

x

For a volume of revolution, we have circular chunks instead of strips.

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Volume of Revolution around x-axis

𝑓 (𝑥)

𝑑𝑥

The volume of one circular chunk is:

So the volume of the whole shape is:∫𝜋 ( 𝑓 (𝑥 ) )2𝑑𝑥

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Around y-axisAround x-axisIntroduction

Volume of Revolution around x-axisExampleLet:

Then on the interval 0 and 1:

xxf )(

∫∫ 1

0

2 .)( xdxdxxvolumeb

a

2)0

2

1(

2

1

0

2 x

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x

𝑓 (𝑥)

Click here to see rotate around the y-axis:

Volume of Revolution around y-axis

Around y-axisAround x-axisIntroduction

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x

y

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x

y

Around y-axisAround x-axisIntroduction

x

y

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Repeat

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𝑦= 𝑓 (𝑥 )

Introduction

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Volume of revolution around y-axis

Volume of revolution around y-axisThis is now the length along the x-axis, so is

This is now the length along the y-axis, so is

Volume of one circle chunk is:

So volume of whole shape is:

∫𝜋 (𝑔 ( 𝑦 ) )2𝑑𝑦

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Volume of revolution around y-axisExample 

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Find the following volumes of revolution:

, from 1 to 5, around x-axis

, from 2 to 4, around y-axis

, from 0 to 3, around y-axis

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∫b

a

dxxf 2))(( ∫b

a

dxyf 2))((

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Check Answers Clear Answers

Show Answers

ConclusionYou should now be able to:

Visualise the effect of rotating a shape around the x and y axes.

Compute the volume of revolution.

Further reading: try looking up the equations needed rotate a shape around the x-axis, this will require knowledge of polar coordinates.

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Introduction