Shell integration
Shell integration (the "
shell method") is a means of
calculating the
volume of a
solid of revolution. This makes use of the so-called "
representative cylinder". 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 a hollow cylinder. Integration, as an accumulative process, can then calculate the
integrated volume of a "
family" of shells (a shell being the outer edge of a hollow cylinder) -- as volume is the
antiderivative of area, if one can calculate the lateral surface area of a shell...one can then calculate its volume.
The necessary equation, for calculating such a volume, V, is slightly different depending on which axis is serving as the axis of revolution. These equations note that the lateral surface area of a shell equals: 2 pi (π) multiplied by the cylinder's average radius, p(y), multiplied by the length of the cylinder, h(y). One can calculate the volume of a representative shell by: 2π * p(y) * h(y) * dy, where dy is the thickness of the shell -- that being some number approaching zero.
- Horizontal Axis of Revolution
- Vertical Axis of Revolution
See also:
disk integration