Shell integration

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Shell integration.svg

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 perpendicular 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.


Mathematically, this method is represented by:

If the rotation is around the y-axis (vertical axis of revolution) then,

2\pi \int_{a}^{b} xh(x)\,dx

where h(x) is the upper function, y = 0 the lower function, a the lower limit, and b the upper limit.

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

2\pi \int_{a}^{b} yh(y)\,dy

where h(y) is the upper function, x = 0 the lower function, a the lower limit, and b the upper limit.


When a region is rotated about any vertical axis, only the circumference part of the integral changes. Instead of being r = x, r = |h -x|, where x= h is the vertical line the region is rotated about. If f(x) > g(x) on [a, b], where x = a and x = b are the lower and upper limits, volume =

2\pi \int_{a}^{b} |h-x|[f(x) - g(x)]\,dx

The same idea applies to rotating a region about any horizontal axis y = h. Make sure to solve each equation for x and change the integration limits to be the y values.

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