ee101 l2 integration volume

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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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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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ee101 l2 integration volume

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