1. VOLUMES BY CYLINDRICAL SHELLS CylinderShellCylinder

2. VOLUMES BY CYLINDRICAL SHELLS Example: Find the volume of the solid obtained by rotating the region enclosed by the curves y=sin(x) and y=0 about the y-axis. Find the volume of the resulting solid. Disk Method Shell Method The method of shells is fundamentally different from the method of disks. The method of disks involves slicing the solid perpendicular to the axis of revolution to obtain the disks. However, the method of shells fills the solid with cylindrical shells This demo was developed by Lila F. Roberts College of Information & Mathematical Sciences Clayton State University Morrow, GA 30260

3. VOLUMES BY CYLINDRICAL SHELLS

4. The volume is given by Find the surface area

5. VOLUMES BY CYLINDRICAL SHELLS step1 Graph and Identify the region Draw a line parallel to the rotating line at the point x step2 Rotate this line about the rotating line step3 Find: in terms of step4 The volume is given by step5 Find: step4 Note: rotating line is y-axis  dx and we draw a parallel line to y-axis

6. VOLUMES BY CYLINDRICAL SHELLS

7. T-122

8. VOLUMES T-102

9. VOLUMES BY CYLINDRICAL SHELLS

10. The volume is given by Find the surface area CYLINDRICAL SHELLS rotating line Parallel to y-axis The volume is given by Find the surface area CYLINDRICAL SHELLS rotating line Parallel to x-axis

11. VOLUMES BY CYLINDRICAL SHELLS Remarks CYLINDRICAL SHELLS (6.2) rotating line Parallel to x-axis rotating line Parallel to y-axis Remarks DESK(6.1) rotating line Parallel to x-axis rotating line Parallel to y-axis Cross-section is DISK Cross—section is WASHER SHELL Method

12. VOLUMES BY CYLINDRICAL SHELLS T-131

13. VOLUMES T-072

14. T-111

15. VOLUMES BY CYLINDRICAL SHELLS T-111