7.3.1 Volume by Disks and Washers
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Transcript of 7.3.1 Volume by Disks and Washers
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7.3.1 Volume by Disks and Washers
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I. Solids of Revolution
A.) Def- If a region in the plane is revolved about a line in the plane, the resulting solid is called a SOLID OF REVOLUTION and the line is called the AXIS OF REVOLUTION.
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B.) Ex. - Find the volume of the solid of revolution generated by rotating the area bounded by
y = 2x, x = 3, and y = 0 about the x-axis.
What does this solid of revolution look like?
6
4
2
-2
2 4 6
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Now, take a slice of the volume by passing two parallel planes perpendicular to the axis of revolution.
6
4
2
-2
2 4 6
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The slice is a circular disk obtained by rotating our representative rectangle about the x-axis.
6
4
2
-2
2 4 6
r
Height or dx
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An approximation of the VOLUME of the SOR
would be .
The actual volume of the SOR can be found by
2
1
2 n
kk
V x x
3
2 2
01 0
lim 2 = 2n
kx
k
V x x x dx
2r
h
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II. Examples Using DisksA.) Ex. 1- Find the volume of the solid of revolution
generated by rotating the area bounded by about the x-axis.225 and 0y x y
4
2
-5 5
2, 25x x
,0x
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4
2
-5 5
2, 25x x
,0x
5 2
2
5
= 25V x dx
5
2
5
= 25V x dx
5
3
5
1 = 25
3V x x
3500 = units
3V
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B.) Ex. 2 - Find the volume of the solid of revolution generated by rotating the area bounded by , x = 0, and y = 4 about the y-axis.
2y x
,y y 0, y
4 2
0
=V y dy4
0
=V ydy4
2
0
1 =
2V y
3 =8 unitsV
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III. Disks – General Case:
A.) Around the x-axis:
B.) Around the y-axis:
2( )
b
a
V f x dx
2( )
d
c
V g y dy
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IV. Washers
ALL RADII MEASURED FROM THE AXIS OF
REVOLUTION. 2 2
2 1V r h r h
r1
r2
2 2
2 1V r r h
2 2
2 1
b
a
V r r dr
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V. Examples Using WashersA.) Ex. 3- Find the volume of the SOR generated by
rotating the area bdd by about the x-axis.
, 1, and 0y x y x
1
,x x
,1x
1 22
0
= 1V x dx
1
0
= 1V x dx 1
2
0
1 =
2V x x
3 = units
2V
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2
1
-1
1 2
B.) Ex. 4- Find the volume of the SOR generated by rotating the area bdd by about the y-axis.
, 1, and 0y x x y
1, y 2 ,y y
1
22 2
0
= 1V y dy
1
4
0
= 1V y dy 1
5
0
1 =
5V y y
34 = units
5V
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VI. Washers– General Case:
A.) Around the x-axis:
B.) Around the y-axis:
2 2( ) ( )
b
a
V f x g x dx
2 2( ) ( )
d
c
V f y g y dy
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VII. Revolving About a Line Other Than the x and y-axes.1.) Find the volume of the SOR generated by rotating the
area bdd by about the line y=5. 2 and 2y x y x 5
4
3
2
1
-1
1 2
2,x x
, 2x x
2
2 22
0
= 5 5 2V x x dx
2
2 4
0
= 20 14V x x x dx 2
2 3 5
0
14 1 = 10
3 5V x x x
3136
= units15
V
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VIII. About x=h or y=k– General Case:
A.) Around the y=k:
B.) Around the x=h:
2 2( ) ( )
b
a
V k f x k g x dx
2 2( ) ( )
d
c
V h f y h g y dy