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www.le.ac.uk
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
Next
Introduction
Examples: click to see the solids formed
Around y-axisAround x-axisIntroduction
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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
Around y-axisAround x-axisIntroduction
Next
x
y
Around y-axisAround x-axisIntroduction
x
Around y-axisAround x-axis
y
Introduction
x
y
Around y-axisAround x-axisIntroduction
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Repeat
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
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
Next
∫ means sum over all the strips
)(xf
)(xf
dx
dx
dx
x
For a volume of revolution, we have circular chunks instead of strips.
Around y-axisAround x-axisIntroduction
Next
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
Next
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
Around y-axisAround x-axisIntroduction
Next
x
𝑓 (𝑥)
Click here to see rotate around the y-axis:
Volume of Revolution around y-axis
Around y-axisAround x-axisIntroduction
Next
x
y
Around y-axisAround x-axisIntroduction
x
y
Around y-axisAround x-axisIntroduction
x
y
Around y-axisAround x-axisIntroduction
Next
Repeat
Around y-axisAround x-axis
𝑦= 𝑓 (𝑥 )
Introduction
Next
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
Around y-axisAround x-axisIntroduction
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Volume of revolution around y-axisExample
Around y-axisAround x-axisIntroduction
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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
Around y-axisAround x-axis
∫b
a
dxxf 2))(( ∫b
a
dxyf 2))((
Introduction
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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.
Around y-axisAround x-axis
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Introduction