Shell integration

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Topics in calculus

Fundamental theorem | Function | Limits of functions | Continuity | Calculus with polynomials | Mean value theorem | Vector calculus | Tensor calculus

Differentiation

Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem | Related rates

Integration

Integration by substitution | Integration by parts | Integration by trigonometric substitution | Solids of revolution | Integration by disks | Integration by cylindrical shells | Improper integrals | Lists of integrals

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

<math>2\pi \int_{a}^{b} p(x) h(x) dx<math>

if the rotation is around the x-axis (horizontal axis of revolution), or

<math>2\pi \int_{a}^{b} p(y) h(y) dy<math>

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

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