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# 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
• V = 2π ∫ p(y)h(y)dy
• Vertical Axis of Revolution
• V = 2π ∫ p(x)h(x)dx