Solid of Revolution (Example: Torus)
1: DISK METHOD
2: WASHERS METHOD
6.2 Volumes by slicing (pg:421)
Volumes by disks and washers perpendicular to the x-axis (page:424)
1. Volumes by Disk Method (pg:424)
6.2.4(p. 424)
Figure 6.2.9(p. 424)
Equation (5)(p. 425)
Curve revolving about x-axis
Example 2: (pg: 425)
Find the volume of the solid that is obtained when the region under the curve
over the interval [1, 4] is revolved about the
x-axis.
xxfy )(
2: Volumes by Washer Method (pg: 426)
Washer Washers
Doughnuts are like Washers
Volumes by washers:1. Perpendicular to the x-axis2. Perpendicular to the y-axis
6.2.5(p. 425)
Figure 6.2.12(p. 425)
Equation (6)(p. 426)
Example 4: (pg: 426)
Find the volume of the solid generated when the region between the graphs of the equations
and g(x)=x over the interval [0, 2] is revolved about the x-axis.
2. Write the formula:
Solution:
1. Draw the graph:
2
2
1)( xxf
Volumes by disks and washers perpendicular to the y-axis (page:426)
Equation (8)
Figure 6.2.14 (p. 427)
Equation (7)
Example 5: (pg: 427)
Find the volume of the solid that is obtained when the region enclosed by the curve
y=2, and x=0 is revolved about the y-axis.
xxfy )(
Figure 6.2.15 (p. 427)
Volumes by washers
b
a
dxrRA )( 22
b
a
dxrRV )( 22
Vwasher = p(R2 – r2)dx
6.3.1(p. 432)
6.3.2(p. 434)
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