ee101 l2 integration volume
TRANSCRIPT
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EE101Calculus and Analytical
Geometry 2
Volumes of Solids of Revolution
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Copyright 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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==
=
=
=
b
a
n
k
kk
n
n
kkk
dxxAxxAV
xxAV
)().(lim
).(
1
1
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Example 1
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Disc Method
~Stack ofPancakes method
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Rotation about x-ax
Example 2
volume.itsFindsolid.a
generatetoaxis-xthe
aboutrevolvedisaxis-x
theand4,x0
,xycurvethebetweenregionThe
=
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Rotation about the horizontal axis
Example 3
Find the volume othe solid generated
by revolving the
region bounded by
and the lines x!1,x!4, about the line
y!1.
xy =
R t ti b t i
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Rotation about y-axis
Example 4
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Rotation about vertical axis
Example 5
Find the volume o thgenerated by revolvin
between the "arabola
and the line x!# abou
x!#.
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Washer Method~Stack of Donuts
method
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Copyright 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
i i
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Rotation about x-axis
Example 6
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Rotation about y-axis
Example 7
l b
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Volumes by
CylindricalShells
~NestedRussian Dolls
method
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rrh
rrhrr
hrrrr
hrr
V
==
+=
+==
=
$
eight).(radius).(haverage.($
)(
$
$
))((
)(
(height).base)oarea(
1$1$
1$1$
$1
$$
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r$ = rhV
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dxxfxV
xxfxV
b
a
)($
)(..$
==
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==b
a
xfLxVLx ()($,lineicalabout vertrevolved&
C li i l h ll l i b t i
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Cylinrical shells revolvin! about y-axis
Example "
Cylinrical shells revolvin! about x axis
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Cylinrical shells revolvin! about x-axis
Example #
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