A Washer
Lingual
- A flat disk, placed beneath a nut or at some joint, to distribute pressure, alleviate friction or prevent leakage.
Synonyms
- (flat disk to distribute pressure or to alleviate friction): blotter
Mathematics
Calculus
The Washer Method (Perpendicular Slices, Hollow Core)
Used when the region between two curves, an outer boundary \(f(x)\) and an inner boundary \(g(x)\), is rotated. The cross-section is a flat ring (washer) with an outer radius \(R\) and inner radius \(r\).
\[\text{Area of cross-section: } A = \pi \left(R^2 - r^2\right)\]
- Rotation about a horizontal axis (\(y = k\)):
\[V = \pi \int_{a}^{b} \left( [R(x)]^2 - [r(x)]^2 \right) dx\]
- Rotation about a vertical axis (\(x = h\)):
\[V = \pi \int_{c}^{d} \left( [R(y)]^2 - [r(y)]^2 \right) dy\]
Common trap: Always subtract the squared radii before integrating: \(\pi \int (R^2 - r^2)\,dx \neq \pi \int (R - r)^2\,dx\).
Comparison of Methods
| Criterion | Disk / Washer Method | Cylindrical Shell Method |
|---|---|---|
| Slice Orientation | Perpendicular (\(\perp\)) to axis of rotation | Parallel (\(\parallel\)) to axis of rotation |
| Differential Element | \(dx\) when axis is horizontal; \(dy\) when axis is vertical | \(dx\) when axis is vertical; \(dy\) when axis is horizontal |
| Typical Formula | \[V = \pi \int (R^2 - r^2) \, d(\text{variable})\] | \[V = 2\pi \int (\text{radius})(\text{height}) \, d(\text{variable})\] |
| Best Used When | Region is easily described as \(y = f(x)\) around horizontal axes, or functions invert cleanly | Expressing the curve in terms of the other variable is difficult or splits into multiple integrals |