Volumes of Revolution
We can calculate the volume of a sphere, cone, or vase by slicing it into infinitely thin disks (like coins) or washers (coins with holes).
The Disk Method is a technique in integral calculus used to find the volume of a solid of revolution. This occurs when you take a two-dimensional region and rotate it around a specific axis (usually the -axis or -axis).
Imagine taking a single thin rectangle from under a curve and spinning it around an axis. It creates a thin, circular disk. By summing up the volumes of an infinite number of these disks using an integral, you find the total volume of the solid.
The Disk Method
How it Works: The Geometry
To understand the formula, think about the volume of a single cylinder (a disk):
In calculus, we represent these parts using the function:
The distance from the axis of rotation to the curve, which is defined by the function .
The "thickness" of the disk, represented by the infinitely small change in , or .
Disk Method Visualizer
Rotate y = √x around the x-axis
The Formula
If you are rotating a continuous function around the -axis from to , the volume is given by:
Key Components:
Pulled outside the integral because it's a constant.
This represents , the squared radius of each disk.
This represents the thickness () of each infinitesimal disk.
Step-by-Step Process
To solve a volume problem using the disk method, follow these steps:
Worked Example
Volume of a Paraboloid
Find the volume of the solid formed by rotating about the -axis from to .
The curve starts at the origin. The shaded area will be rotated.
Rotating about the -axis → use . Radius .
From to (marked in red).
Level Up Examples
Example A: The Washer Method
Find the volume when the region between and is rotated about the -axis.
Example B: Shifted Axis of Rotation
Find the volume when from to is rotated about .
Example C: Rotating About the Y-Axis
Find the volume when (equivalently ) from to is rotated about the -axis.
Practice Quiz
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