1. Cylinder Cylinder Shell
VOLUMES BY CYLINDRICAL SHELLS
Cylinder
Cylinder
Shell

2. VOLUMES BY CYLINDRICAL SHELLS

3. E-II T-061 VOLUMES BY CYLINDRICAL SHELLS step1 step2 step3 step4 step4
Graph and Identify the region
Draw a line parallel to the rotating line at the point x
step2
step3
Rotate this line about the rotating line
step4
Find:
in terms of
step4
Find:
step5
The volume is given by
Note: rotating line is y-axis  dx and we draw a parallel line to y-axis

4. VOLUMES
T-141

5. T-122 VOLUMES BY CYLINDRICAL SHELLS 1 2 3 4 5 6
Graph and Identify the region
Draw a line parallel to the rotating line at the point y 2 3
Rotate this line about the rotating line 4
Find:
in terms of 5
Find: 6
The volume is given by
Note: rotating line is x-axis  dy and we draw a parallel line to x-axis

6. VOLUMES BY CYLINDRICAL SHELLS

7. VOLUMES
SUMMARY:
The solids in all previous examples are all called solids of revolution because they are obtained by revolving a region about a line.
DISK
SHELLS
WASHER
Rotated by x-axis or a line parallel to x-axis
Rotated by y-axis or a line parallel to y-axis

9. Final T-131

10. VOLUMES
T-102

11. VOLUMES
T-131