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Shell integration (the shell method in integral calculus) is a means of calculating the
volume of a solid of
revolution.
It makes use of the so-called "representative cylinder". Intuitively speaking,
part of the graph of a function is rotated around an axis, and is modelled by an infinite number of hollow pipes, all infinitely thin.
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.
Mathematically, take
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if the rotation is around the x-axis (horizontal axis of revolution), or
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if the rotation is around the y-axis (vertical axis of revolution).
So here the function p(.) is the distance from the axis and h(.) is generally the function being rotated.
The values for a and b are the limits of integration, the starting and stopping points of the rotated shape
(i.e. the points delimiting the section of the graph we use).
See also: disk integration, Disc Method
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