Disc integration
Disk integration (the "disk method") is a means of calculating the volume of a solid of revolution. This makes use of the so-called "representative disk". The idea is that a "representative rectangle" (used in the most basic forms of integration -- such as ∫ x dx) can be rotated about the axis of revolution; thus generating such a disk.
- Horizontal Axis of Revolution
- V = π ∫ [R(x)]2 dy
- Vertical Axis of Revolution
- V = π ∫ [R(y)]2 dy
These equations note that the area of a disk equals pi, π, multiplied by the disk's radius. As volume is the antiderivative of area, the integral can be used to find a volume formed by a "family" of disks.
For instance, consider the function f(x) = √(sin x), as it exists between x = 0 and x = π. If one imagines this function being rotated around the x-axis (soas to create a solid of revolution); then, the radius of that solid (for any value, x) is equal to √(sin x). Using the above formula, we can determine the solid to have a volume of: π ∫ [√(sin x)]2 dx -- when evaluated from 0 to π. The solid has a volume of 2π.