Volume by Washers (Solid of Revolution)

Definition

The washer method calculates the volume of a solid of revolution when the region being revolved does not border the axis of revolution. The volume $V$ is: $V = \int_{a}^{b} \pi \left( [R(x)]^2 - [r(x)]^2 \right) \, dx$ where $R(x)$ is the outer radius and $r(x)$ is the inner radius.

• How to read: “V equals integral from a to b of pi times (R of x squared minus r of x squared) dx.”
• Meaning: Annulus area $\pi(R^2 - r^2)$ at each slice; outer minus inner disk when the axis lies outside the region.

Why It Matters

Most real-world mechanical parts (like pipes and gears) aren’t solid; they have holes. The washer method provides the mathematical precision to account for this “emptiness,” ensuring that we don’t over-order materials or miscalculate the weight of hollow structures.

Core Concepts

• Washer Geometry: The cross-section is an annulus (washer) with area $A = \pi(\text{outer radius})^2 - \pi(\text{inner radius})^2$.
  • How to read: “A equals pi times outer radius squared minus pi times inner radius squared.”
  • Meaning / when to use: Hollow cross-section; subtract inner disk from outer disk.
• Hollow Centers: This method accounts for the “hole” created when a region is rotated at a distance from the axis.
• Radial Distance: $R(x)$ and $r(x)$ are the distances from the axis of revolution to the outer and inner boundaries, respectively.

Connected Concepts

• Volume by Cylindrical Shells
• Volume by Disks (Solid of Revolution)
• The LIATE Rule