6.2 volume of solid of revolution

Volumes of solids of revolution by Disk and Washer method with examples.

1. 6.2Volumes of solids of revolution

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.

5. Solid of Revolution (Example: Torus)

6. 1: Disk method 2: Washers Method 6.2 Volumes by slicing (pg:421)

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

8. 6.2.4 (p. 424) Figure 6.2.9 (p. 424) Equation (5) (p. 425)

9. About x-axis   .  

10. 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.

11. 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 = e x , 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

12. 2: Volumes by Washer Method (pg: 426) Washer Washers

13. Doughnuts are like Washers

14. Volumes by washers: Perpendicular to the x-axis Perpendicular to the y-axis

15. 6.2.5 (p. 425) Figure 6.2.12 (p. 425) Equation (6) (p. 426)

16. 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.

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

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

19. 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.

20. Figure 6.2.15 (p. 427)

21. Volumes by washers V washer = p( R 2 – r 2 ) dx

22. 6.3.1 (p. 432) 6.3.2 (p. 434)

23. http://mathdemos.org/mathdemos/washermethod/gallery/gallery.html