section 7-2 solids of revolution disk method. the disk method if a region in the plane is revolved...
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The Disk Method
If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution.
Figure 7.13
The simplest such solid is a rightcircular cylinder or disk, which isformed by revolving a rectangleabout an axis adjacent to oneside of the rectangle
To see how to use the volume of a disk to find the volume of a general solid of revolution, consider a solid of revolution formed by revolving the plane region in Figure 7.14 about the indicated axis.
Figure 7.14
The Disk Method
How do you find the Volume of a solid generated by revolving a given area about an axis?
Slice the volume into many, many circular disks
Then add up the volume of all the disks
Solids of Revolution: Disk Method• The volume of a solid may be found by finding
the sum of the disks. • The volume of each circular disk is the area of a
circle times the width of the disk.
• Volume is found by integration. The radius of each disk is the function for each value in the interval. The width is dx
b
a
dxxfV 2))((
wxRn
1
2))((
Find Volume using Disk Method
b
a
dxxfV 2))(( d
c
dyyfV 2))((
• Revolve about a horizontal axis
• Slice perpendicular to axis – slices vertical
• Integrate in terms of x
• Revolve about a vertical axis
• Slice perpendicular to axis – slices horizontal
• Integrate in terms of y
Find Volume of a solid generated by revolving the given area about the x-axis1) Consider the function
on the interval [0,2]xxf )(
A xdx0
2
1) (Continued) Find the volume of the solid bounded by and the x-axis rotated about the x-axis on the interval [0,2]
V x 2
dx0
2
2
0
xdxV
xxf )(
2
0
2
2
xV
2)02( V
2) Find the volume of the solid generated by revolving the region bounded by y = x – x2 and y = 0 about the x - axis
1
.25
1
0
543
523
xxx
V
1
0
2dxxxV 2
1
0
432 2 dxxxxV
305
1
2
1
3
1 543
V
3) Find the volume generated by revolving the
region bounded by y = sec(x), and
y = 0 about the x - axis
2
)0tan(4tan2
V
4,
4
xx
40))(tan(2
xV
4
4
2))(sec(
dxxV
4
0
2 )(sec2
dxxV
4
4
4) Find the volume generated by revolving the
region bounded by about
the y - axis
4
81V
23yx
3
0
22
3dyyV
3 and ,0,32
yxxy
3
0
4
4
yV
Need in terms of x = ? Since revolution is about y
3
0
3dyyV
The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of the following equation revolved about the y-axis:
2.000574 .439 185x y y x
y
500 ft
500 22
0.000574 .439 185 y y dy
5) Find the volume using the disk method with a horizontal disk.
V=
24,732,567 ft 3