Disk Method — Deriving the Cone Volume

Calculus 1 · Axiom Academy

EXAMPLE Volume of Revolution - Disk Method Deriving the volume of a cone with calculus by stacking thin circular disks Take the line on the interval and rotate it about the x -axis. This sweeps out a cone with base radius 2 and height 3 . Use the disk method to find its volume V — and check that the answer matches the familiar cone formula. Each cross-section perpendicular to the x -axis is a circular disk of radius . The base disk at x = 3 has radius 2 ; the highlighted disk at x = 2 has radius . Nice work — you derived the cone-volume formula from scratch using the disk method. Linear radius: a line through the origin gives for a cone of base radius R and height h . Disk volume element: each thin slice is a disk of volume , and the integral stacks them all up. Classic result: — exactly the geometric cone formula. The same recipe — write the radius, square it, integrate — finds the volume of any solid of revolution.

This is the written version of the interactive lesson above. See the full Calculus 1 course.