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| [[File:Shell integration.svg|thumb|right|200px]]
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| {{Calculus |Integral}}
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| '''Shell integration''' (the '''shell method''' in [[integral calculus]]) is a means of [[calculation|calculating]] the [[volume]] of a [[solid of revolution]], when integrating along an axis ''perpendicular to'' the axis of revolution. While less intuitive than [[disc integration]], it usually produces simpler integrals.
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| It makes use of the so-called "representative [[cylinder (geometry)|cylinder]]". Intuitively speaking, part of the [[graph of a function]] is rotated around an [[axis of rotation|axis]], and is modelled by an infinite number of hollow pipes, all infinitely thin.
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| The idea is that a "representative [[rectangle]]" (used in the most basic forms of [[integral|integration]] – such as ∫ ''x'' ''dx'') can be rotated about the [[solid of revolution|axis of revolution]]; thus generating a hollow cylinder. Integration, as an accumulative process, can then calculate the ''integrated'' volume of a "[[Family (disambiguation)#Mathematics|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.
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| Shell integration can be considered a special case of evaluating a [[double integral]] in [[polar coordinates]].
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| == Calculation ==
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| Mathematically, this method is represented by:
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| if the rotation is around the [[y-axis]] (vertical axis of revolution) then,
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| :<math>2\pi \int_{a}^{b} xh(x)\,dx</math> | |
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| else if the rotation is around the [[x-axis]] (horizontal axis of revolution) then,
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| :<math>2\pi \int_{a}^{b} yh(y)\,dy</math>
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| [[File:Shell-integration.png|thumb|center|200px]]
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| ==See also==
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| *[[Solid of revolution]]
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| *[[Disk integration]]
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| *[[Radius]]
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| ==References==
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| *[http://www.cliffsnotes.com/study_guide/topicArticleId-39909,articleId-39907.html CliffsNotes.com. Volumes of Solids of Revolution]. 12 Apr 2011.
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| *{{MathWorld|title=Method of Shells|urlname=MethodofShells}}
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| *Frank Ayres, Elliott Mendelson:''Schaum's outlines: Calculus''. McGraw-Hill Professional 2008, ISBN 978-0-07-150861-2. pp. 244–248 ({{Google books|Ag26M8TII6oC|online copy|page=244}})
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| {{DEFAULTSORT:Shell Integration}}
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| [[Category:Integral calculus]]
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