Volume of Revolution - Disk Method

Calculus 1 · Axiom Academy

EXAMPLE Volume of Revolution — Disk Method Rotating about the x -axis and adding up thin circular disks. The region under from x=0 to x=4 is rotated about the x -axis, sweeping out a solid. Find its volume using the disk method. Spin the orange region around the x -axis. A vertical slice at position x becomes a thin disk whose radius is the curve height . Stack the disk areas from x=0 to x=4 to get the volume. Nice work — you built a solid of revolution one disk at a time. The pieces worth keeping: Disk-method formula: , where R(x) is the distance from the axis to the curve. Radius from the curve: rotating about the x -axis makes the radius the height of the graph, . Square before you integrate: collapses an awkward square root into a tidy . Same idea, different radius: whenever a region is spun about an axis it touches, slice it into disks of area and integrate along the axis of rotation.

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