Volumes of solids of revolution. Examples with animations (from

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