1. The document discusses different methods for calculating the volume of solids obtained by revolving a region about an axis, including the disk method, washer method, and examples applying each method. 2. The disk method is used when the region is revolved about the x-axis, and the washer method is used when the region is revolved about the y-axis. 3. Examples are provided to demonstrate calculating the volume of solids generated by revolving regions between curves over intervals about the x-axis and y-axis.

2. Right Circular Cylinders

3. Volume of Right Circular Cylinders

4. Example 3: pg: 425
Derive the formula for the volume of a sphere of
radius r.

6. Solid of Revolution (Example: Torus)

11. 1: DISK METHOD
2: WASHERS METHOD
6.2 Volumes by slicing (pg:421)

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

13. 6.2.4
(p. 424)
Figure 6.2.9
(p. 424)
Equation
(5)
(p. 425)

14. About x-axis

15. 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  )(

16. 11–18 Find the volume of the solid that results when the
region
enclosed by the given curves is revolved about the x-
axis. "
11. y = '25 − x2, y = 3
12. y = 9 − x2, y = 0
13. x = 'y, x = y/4
14. y = sin x, y = cos x, x = 0, x = "/4
[Hint: Use the identity cos 2x = cos2 x − sin2 x.]
15. y = ex, y = 0, x = 0, x = ln 3
16. y = e−2x, y = 0, x = 0, x = 1
17. y =1'4 + x2
, x = −2, x = 2, y = 0
18. y =
e3x
'1 + e6x
, x = 0, x = 1, y = 0

19. 2: Volumes by Washer Method (pg: 426)
 Washer  Washers

20. Doughnuts are like Washers

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

23. 6.2.5
(p. 425)
Figure 6.2.12
(p. 425)
Equation (6)
(p. 426)

24. 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
2
1
)( xxf 

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

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

29. 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  )(

32. Figure 6.2.15 (p. 427)

34. Volumes by washers
 
b
a
dxrRA )( 22
 
b
a
dxrRV )( 22

Vwasher = p(R2 – r2)dx

40. 6.3.1
(p. 432)
6.3.2
(p. 434)

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