The document discusses methods for calculating the volumes of solids obtained by revolving a region about an axis. It covers the disk method, washer method, and examples of revolving regions about the x-axis and y-axis. Regions are revolved between curves and axes over intervals to obtain solids, and formulas are provided for calculating volumes based on disks, washers, or cylindrical shells. Examples demonstrate setting up and solving problems to find volumes of solids using these methods.

2. Solid of Revolution (Example: Torus)

7. 6.2 Volumes by slicing (pg:421)
1: DISK METHOD
2: WASHERS METHOD

8. Volumes by disks and washers
perpendicular to the x-axis (page:424)

9. 1. Volumes by Disk Method (pg:424)

10. 6.2.4
(p. 424)
Figure 6.2.9
(p. 424)
Equation
(5)
(p. 425)

11. Curve revolving about x-axis

12. Example 2: (pg: 425)
 Find the volume of the
solid that is obtained
when the region under
the curve
y  f (x)  x
over the interval [1, 4]
is revolved about the
x-axis.

16. 2: Volumes by Washer Method (pg: 426)
 Washer  Washers

17. Doughnuts are like Washers

19. Volumes by washers:
1. Perpendicular to the x-axis
2. Perpendicular to the y-axis

20. 6.2.5
(p. 425)
Figure 6.2.12
(p. 425)
Equation (6)
(p. 426)

21. Example 4: (pg: 426)
 Find the volume of the
solid generated when
the region between the
graphs of the equations
1
f (x)   x
and g(x)=x over the
interval [0, 2] is
revolved about the x-axis.
 Solution:
 1. Draw the graph:
 2. Write the formula:
2
2

22. Volumes by disks and washers
perpendicular to the y-axis (page:426)

23. Equation
(8)
Equation (7)
Figure 6.2.14 (p. 427)

26. Example 5: (pg: 427)
 Find the volume of the
solid that is obtained
when the region
enclosed by the curve
y  f (x)  x
y=2, and x=0 is
revolved about the y-axis.

29. Figure 6.2.15 (p. 427)

31. Volumes by washers
b
A ( R r ) dx 2 2   
  
V ( R r ) dx 2 2  
a
b
a
Vwasher = p(R2 – r2)dx

38. 6.3.1
(p. 432)
6.3.2
(p. 434)

40. http://mathdemos.org/mathdemos/washe
rmethod/gallery/gallery.html