← Unit 8: Applications of Integration
Volumes of Revolution: Disk and Washer Methods
Rotating a region around an axis creates a solid whose volume can be found by integrating cross-sectional areas — circles (disk method) when the region touches the axis, or rings (washer method) when it doesn't.
Disk and washer methods
Disk method (region bounded by curve and the axis of rotation): V = π∫[a,b] [R(x)]² dx, where R(x) is the distance from the axis to the curve.
Washer method (region between two curves, neither touching the axis): V = π∫[a,b] ([R(x)]² - [r(x)]²) dx, where R is the outer radius and r is the inner radius.
Worked Example
Find the volume when the region between y=√x and the x-axis on [0,4] is rotated about the x-axis.
- Since the region touches the axis (y=0), use the disk method with R(x)=√x.
- V = π∫[0,4] (√x)² dx = π∫[0,4] x dx = π[x²/2] from 0 to 4
- = π(16/2 - 0) = 8π
Answer: 8π
Key terms (2)
- Disk method
- V = π∫[R(x)]²dx, for a region rotated about an axis it touches.
- Washer method
- V = π∫([R(x)]²-[r(x)]²)dx, for a region rotated about an axis it does not touch.
Practice Quiz
Question 1 of 2Score so far: 0/0
When should you use the washer method instead of the disk method?