1. Volumes by Disks and Washers Or, how much toilet paper fits on one of those huge rolls, anyway??

2. Volume by Slicing Approximating area  rectangles Volume = length x width x height Total volume =  (A x  t) Volume of a slice = Area of a slice x Thickness of a slice A tt

3. Volume by Slicing Total volume =  (A x  t) VOLUME =  A dt But as we let the slices get infinitely thin, Volume = lim  (A x  t)  t  0 Recall: A = area of a slice

4. Rotating a Function Such a rotation traces out a solid shape (in this case, we get something like half an egg)

5. Volume by Slices Slice r dt

6. Disk Formula VOLUME =  A dt

7. Volume by Disks (Rotation about the x – axis) r thickness x axis y axis Slice radius

8. Example

9. Homework: p. 324 #1-4 *Sketch the region first!

10. Example

11. More Volumes f(x) g(x) rotate around x axis Slice R r Area of big circle – hole

12. Washer Formula VOLUME =  A dt

13. Volumes by Washers (about the x – axis) f(x) g(x) Slice R r dt Big R little r dx

14. 2 The application we’ve been waiting for... 1 rotate around x axis 1 0.5 f(x) g(x)

15. Toilet Paper f(x) g(x) 1 2 0.5 1 So we see that: f(x) = 2, g(x) = 0.5 0 x only goes from 0 to 1, so we use these as the limits of integration. Now, plugging in our values for f and g:

16. Example Homework: p. 324 #5, 6, 8, 9 *Sketch the region first!

17. Example

18. (horizontal)

19. Volume by Cross Sections Recall slices: VOLUME =  A dt A = area of cross sections of the figure  need area of common shapes Use x when cross-sections are perpendicular to the x – axis. Use y when perpendicular to the y – axis.

20. Example base Homework: handout #55, 56

21. TI-84 Calculator Methods

23. Test – Area and Volume