Shell integration

Topics in Calculus

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Shell integration (the shell method in integral calculus) is a means of calculating the volume of Newns when integrating along an axis perpendicular to the axis 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 π multiplied by the cylinder's average radius, p(x), multiplied by the height of the cylinder, h(x). One can calculate the volume of a representative shell by: 2π * p(x) * h(x) * dx, where dx is the thickness of the shell – that being some number approaching zero.

Shell integration can be considered a special case of evaluating a double integral in polar coordinates.

Calculation

Mathematically, this method is represented by:

$2\pi \int_{a}^{b} (y)(p(y)-h(y))\,dy$

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

$2\pi \int_{a}^{b} (x)(p(x)-h(x))\,dx$

if the rotation is around the y-axis (vertical axis of revolution).

So here the function p(x) is the distance from the axis and h(x) is the length of the shell, 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 (note the limits are units of the Axis of Revolution).

See also

• Solid of revolution
• Disk integration
• Radius

The surface area of Cylinder by double Integration, derived from following equation: S=∫_0^l▒∫_0^r▒〖2π r dr dθ〗 And we know that, the Length of arc is: z=ab=rθ→dz=r dθ In conclusion, the surface area is: =>S=∫_0^l▒∫_0^r▒〖2π r dr dθ=2π〗 ∫_0^l▒〖rdz=2πrl〗

References

• CliffsNotes.com. Volumes of Solids of Revolution. 12 Apr 2011 <http://www.cliffsnotes.com/study_guide/topicArticleId-39909,articleId-39907.html>.
• Weisstein, Eric W., "Method of Shells" from MathWorld.
• Frank Ayres, Elliott Mendelson:Schaum's outlines: Calculus. McGraw-Hill Professional 2008, ISBN 9780071508612. pp. 244-248 (online copy at Google Books)