Volume by Disks (Solid of Revolution)

Definition

A solid of revolution is generated by rotating a planar region about an axis. If the region borders the axis and cross-sections are circular disks, the volume $V$ is: $V = \int_{a}^{b} \pi [R(x)]^2 \, dx$ where $R(x)$ is the radius function.

• How to read: “V equals integral from a to b of pi times R of x squared dx.”
• Meaning: Sum disk areas $\pi r^2$ where $r = R(x)$; each slice perpendicular to the $x$-axis is a solid disk.

Why It Matters

Objects of revolution—from pistons to planets—are ubiquitous in engineering. The disk method is the fundamental tool for calculating the “mass and space” of these objects; without it, we couldn’t design even the simplest mechanical engine component.

Core Concepts

• Disk Geometry: Each cross-section perpendicular to the axis of revolution is a circular disk with area $A = \pi r^2$.
  • How to read: “Disk area A equals pi r squared.”
  • Meaning: Cross-section is a full circle; radius is distance from axis to boundary.
• Radius Function: The radius $r$ is the distance from the axis of revolution to the boundary of the region, expressed as $R(x)$ or $R(y)$.
• Axis of Revolution: The variable of integration corresponds to the axis about which the region is rotated.

Connected Concepts

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