Volumes of Revolution - Complete Interactive Lesson
Part 1: Disk Method
Volumes of Revolution
Part 1 of 7 โ Disk Method
Rotating Around the x-axis
When rotating around the -axis, each cross-section is a disk with radius :
Worked Example
Find the volume when from to is rotated about the -axis.
Disk Method ๐ฏ
Key Takeaways โ Part 1
- Disk method:
Part 2: Washer Method
Volumes of Revolution
Part 2 of 7 โ Washer Method
When There's a Hole
Rotating a region between two curves creates a washer (disk with a hole):
Part 3: Rotation About Other Axes
Volumes of Revolution
Part 3 of 7 โ Rotation About Other Axes
Rotating About or
When rotating about a line other than the -axis, adjust the radii:
Rotation about (horizontal line):
Part 4: Cross-Sectional Volumes
Volumes of Revolution
Part 4 of 7 โ Cross-Sectional Volumes
Known Cross-Sections
Instead of rotating, we can have cross-sections that are squares, semicircles, equilateral triangles, etc.
where is the area of the cross-section at position .
Part 5: Disk/Washer in y
Volumes of Revolution
Part 5 of 7 โ Disk/Washer in
Rotating About the -axis Using
Part 6: AP-Style Workshop
Volumes of Revolution
Part 6 of 7 โ AP-Style Workshop
AP-Style Volume Problems ๐ฏ
Workshop Complete!
Part 7: Comprehensive Assessment
Volumes of Revolution โ Review
Part 7 of 7 โ Comprehensive Assessment
| Method | Formula |
|---|---|
| Disk | |
| Washer |