2. -

3. Volumes of a Solid The volumes of solid that can be cut into thin slices, where the volumes can be interpreted as a definite integral.

4. Slicing the solid into thin slabs

5. The formula of volume of Solid The volume V of the solid should be given approximately by the Riemann sum

6. Solids of Revolution 1. The Method of Disks 2. The Method of Shells

7. Example Let R be the regions bounded by the graphs of and Compute the volume of the solid formed by revolving R about y-axis

8. Revolving about y-axis

9. The Method of Disks

10. The Method of Disks (2)

11. Revolving about x-axis

12. Revolving about x-axis (animate)

13. The Solid by revolving about x-axis

15. Example Let R be the regions bounded by the graphs of and Compute the volume of the solid formed by revolving R about y-axis

16. Example(continued)

17. Example Let R be the regions bounded by the graphs of and Compute the volume of the solid formed by revolving R about x-axis

18. Example(continued)