Shell integration

Calculus
Fundamental theorem Limits of functions Continuity Mean value theorem Rolle's theorem
Differential calculus Definitions Derivative Differential (infinitesimal) Differential of a function Generalizations of the derivative Concepts Notation for differentiation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem Rules and identities Sums Products Chain rule Powers Quotients Faà di Bruno's formula
Integral calculus Lists of integrals Definitions Antiderivative Integral Improper integral Riemann integral Lebesgue integration Contour integral Integration by parts discs cylindrical shells substitution trigonometric substitution partial fractions changing order reduction formulae
Series Geometric Arithmetico-geometric Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (Term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector calculus Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem Directional derivative
Multivariable calculus Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian
Specialized calculi Fractional Stochastic Calculus of variations

Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution, when integrating along an axis parallel to the axis of revolution. While less intuitive than disc integration, it usually produces simpler integrals.

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.

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

Calculation

Mathematically, this method is represented by:

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

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

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

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

See also

• Solid of revolution
• Disk integration
• Radius

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", MathWorld.
• Frank Ayres, Elliott Mendelson:Schaum's outlines: Calculus. McGraw-Hill Professional 2008, ISBN 978-0-07-150861-2. pp. 244–248 (online copy at Google Books)