www.le.ac.uk integration – volumes of revolution department of mathematics university of leicester
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www.le.ac.uk
Integration – Volumes of revolution
Department of MathematicsUniversity of Leicester
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Content
Around y-axis
Around x-axis
Introduction
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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
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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
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y
x
𝑓 (𝑥)
Click here to see rotate around the x-axis:
Volume of Revolution around x-axis
Around y-axisAround x-axisIntroduction
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x
y
Around y-axisAround x-axisIntroduction
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x
Around y-axisAround x-axis
y
Introduction
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x
y
Around y-axisAround x-axisIntroduction
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Repeat
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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
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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
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Around y-axisAround x-axis
Another way of looking at integration
Introduction
Next
∫ means sum over all the strips
)(xf
)(xf
dx
dx
dx
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x
For a volume of revolution, we have circular chunks instead of strips.
Around y-axisAround x-axisIntroduction
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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
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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
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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
Next
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x
y
Around y-axisAround x-axisIntroduction
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x
y
Around y-axisAround x-axisIntroduction
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x
y
Around y-axisAround x-axisIntroduction
Next
Repeat
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Around y-axisAround x-axis
𝑦= 𝑓 (𝑥 )
Introduction
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Volume of revolution around y-axis
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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
Around y-axisAround x-axisIntroduction
Next
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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
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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
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