Step by Step
Disk
The Disk method
Used when a solid region (with no hole) is rotated around an axis. The volume is V = π∫f(x)²dx — treating each cross-section as a solid circular disk.
Washer
The Washer method
Used when the region between two curves is rotated around an axis, creating a hole through the middle. The volume is V = π∫(outer² − inner²)dx — subtracting the inner "missing" disk from the outer one, like a washer.
Shell
The Shell method
Used especially when rotating around the y-axis is awkward with disks/washers. The volume is V = 2π∫x·f(x)dx — treating the solid as a series of thin cylindrical shells stacked inside each other.
Pick
Choosing which method to use
Disk and washer integrate perpendicular to the axis of rotation; shell integrates parallel to it. All three methods give the same final volume for a given solid — choose whichever produces the simpler integral to evaluate for a particular problem.
Applied Walkthrough
1
Find the volume when the region under f(x) = x² from x=0 to x=2 is rotated around the x-axis. Since there's no gap or hole, use the disk method.
2
Apply the disk formula: V = π∫₀² (x²)² dx = π∫₀² x⁴ dx.
3
Integrate: π[x⁵/5] from 0 to 2 = π(32/5 − 0) = 32π/5.
4
If instead the region between two curves (say f(x) = x² and g(x) = x) were rotated, creating a gap between the two curves, the washer method would be needed instead, subtracting the inner curve's contribution from the outer curve's.
Exam Application
Exams test whether you can identify which method (disk, washer, or shell) fits a given rotation setup, and whether you can correctly set up and evaluate the resulting integral.
⚠ Common Trap
The most common trap is using the disk method when there's actually a gap between two curves (which requires the washer method instead) — forgetting to subtract the inner curve's contribution gives a volume that's too large.
✓ Quick Self-Check
1. What is the disk method formula?
V = π∫f(x)²dx.
Tap to reveal / hide
2. When do you need the washer method instead of the disk method?
When the region being rotated is between two curves, creating a hole through the middle of the solid.
Tap to reveal / hide
3. What is the shell method formula?
V = 2π∫x·f(x)dx.
Tap to reveal / hide
4. Find the volume when the region under f(x) = x² from x=0 to x=2 is rotated around the x-axis.
32π/5.
Tap to reveal / hide
5. How do disk/washer and shell methods differ in their orientation relative to the axis of rotation?
Disk and washer integrate perpendicular to the axis of rotation; shell integrates parallel to it.
Tap to reveal / hide