Editing Solid of revolution (section)

==Finding the volume==
Two common methods for finding the volume of a solid of revolution are the [[Disc integration|disc method]] and the [[Shell integration|shell method of integration]]. To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness {{mvar|δx}}, or a cylindrical shell of width {{mvar|δx}}; and then find the limiting sum of these volumes as {{mvar|δx}} approaches 0, a value which may be found by evaluating a suitable integral. A more rigorous justification can be given by attempting to evaluate a [[triple integral]] in [[cylindrical coordinates]] with two different orders of integration.

===Disc method===
[[File:Disc integration.svg|thumb|right|Disc integration about the y-axis]]
{{main|Disc integration}}

The disc method is used when the slice that was drawn is ''perpendicular to'' the axis of revolution; i.e. when integrating ''parallel to'' the axis of revolution.

The volume of the solid formed by rotating the area between the curves of {{math|''f''(''y'')}} and {{math|''g''(''y'')}} and the lines {{math|1=''y'' = ''a''}} and {{math|1=''y'' = ''b''}} about the {{mvar|y}}-axis is given by
<math display="block">V = \pi \int_a^b \left| f(y)^2 - g(y)^2\right|\,dy\, .</math>
If {{math|1=''g''(''y'') = 0}} (e.g. revolving an area between the curve and the {{mvar|y}}-axis), this reduces to:
<math display="block">V = \pi \int_a^b f(y)^2 \,dy\, .</math>

The method can be visualized by considering a thin horizontal rectangle at {{mvar|y}} between {{math|''f''(''y'')}} on top and {{math|''g''(''y'')}} on the bottom, and revolving it about the {{mvar|y}}-axis; it forms a ring (or disc in the case that {{math|1=''g''(''y'') = 0}}), with outer radius {{math|''f''(''y'')}} and inner radius {{math|''g''(''y'')}}.  The area of a ring is {{math|π(''R''<sup>2</sup> − ''r''<sup>2</sup>)}}, where {{mvar|R}} is the outer radius (in this case {{math|''f''(''y'')}}), and {{mvar|r}} is the inner radius (in this case {{math|''g''(''y'')}}). The volume of each infinitesimal disc is therefore {{math|π''f''(''y'')<sup>2</sup> ''dy''}}. The limit of the Riemann sum of the volumes of the discs between {{mvar|a}} and {{mvar|b}} becomes integral (1).

Assuming the applicability of [[Fubini's theorem]] and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D):
<math display="block">V = \iiint_D dV = \int_a^b \int_{g(z)}^{f(z)} \int_0^{2\pi} r\,d\theta\,dr\,dz = 2\pi \int_a^b\int_{g(z)}^{f(z)} r\,dr\,dz = 2\pi \int_a^b \frac{1}{2}r^2\Vert^{f(z)}_{g(z)} \,dz = \pi \int_a^b (f(z)^2 - g(z)^2)\,dz</math>

===Shell Method of Integration ===
{{main|Shell integration}}
[[File:Shell integration.svg|thumb|right|Shell integration]]

The shell method (sometimes referred to as the "cylinder method") is used when the slice that was drawn is ''parallel to'' the axis of revolution; i.e. when integrating ''perpendicular to'' the axis of revolution.

The volume of the solid formed by rotating the area between the curves of {{math|''f''(''x'')}} and {{math|''g''(''x'')}} and the lines {{math|1=''x'' = ''a''}} and {{math|1=''x'' = ''b''}} about the {{mvar|y}}-axis is given by
<math display="block">V = 2\pi \int_a^b x |f(x) - g(x)|\, dx\, .</math>
If {{math|1=''g''(''x'') = 0}} (e.g. revolving an area between curve and {{mvar|x}}-axis), this reduces to:
<math display="block">V = 2\pi \int_a^b x | f(x) | \,dx\, .</math>

The method can be visualized by considering a thin vertical rectangle at {{mvar|x}} with height {{math|''f''(''x'') − ''g''(''x'')}}, and revolving it about the {{mvar|y}}-axis; it forms a cylindrical shell.  The lateral surface area of a cylinder is {{math|2π''rh''}}, where {{mvar|r}} is the radius (in this case {{mvar|x}}), and {{mvar|h}} is the height (in this case {{math|''f''(''x'') − ''g''(''x'')}}).  Summing up all of the surface areas along the interval gives the total volume.

This method may be derived with the same triple integral, this time with a different order of integration:
<math display="block">V = \iiint_D dV = \int_a^b \int_{g(r)}^{f(r)} \int_0^{2\pi} r\,d\theta\,dz\,dr = 2\pi \int_a^b\int_{g(r)}^{f(r)} r\,dz\,dr = 2\pi\int_a^b r(f(r) - g(r))\,dr.</math>

{{multiple image
 | align = center
 | direction = horizontal
 | width = 500
 | header = Solid of revolution demonstration
 | image1 = Revolução de poliedros 01.jpg
 | alt1 = five coloured polyhedra mounted on vertical axes
 | caption1 = The shapes at rest
 | image2 = Revolução de poliedros 02.jpg
 | alt2 = five solids of rotation formed by rotating polyhedra
 | caption2 = The shapes in motion, showing the solids of revolution formed by each
}}