Editing Solid of revolution

{{Short description|Type of three-dimensional shape}}
[[File:Rotationskoerper animation.gif|thumb|right|Rotating a curve. The surface formed is a [[surface of revolution]]; it encloses a solid of revolution.]]
[[File:Revolução de poliedros 03.webm|thumb|Solids of revolution ([[Matemateca|Matemateca Ime-Usp]])]]

In [[geometry]], a '''solid of revolution''' is a [[Solid geometry|solid figure]] obtained by [[rotating]] a [[plane figure]] around some [[straight line]] (the ''[[axis of revolution]]''), which may not [[Intersection (geometry)|intersect]] the [[generatrix]] (except at its boundary). The [[Surface (mathematics)|surface]] created by this revolution and which bounds the solid is the ''[[surface of revolution]]''.

Assuming that the curve does not cross the axis, the solid's [[volume]] is equal to the [[length]] of the [[circle]] described by the figure's [[centroid]] multiplied by the figure's [[area]] ([[Pappus's centroid theorem|Pappus's second centroid theorem]]).

A '''representative disc''' is a three-[[dimension]]al [[volume element]] of a solid of revolution.  The element is created by rotating a [[line segment]] (of [[length]] {{mvar|w}}) around some axis (located {{mvar|r}} units away), so that a [[cylinder (geometry)|cylindrical]] [[volume]] of {{math|π''r''<sup>2</sup>''w''}} units is enclosed.

==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
}}

==Parametric form==
[[File:Paolo uccello, studio di vaso in prospettiva 02.jpg|thumb|[[Mathematics and art]]: study of a vase as a solid of revolution by [[Paolo Uccello]]. 15th century]]

When a curve is defined by its [[Parametric equation|parametric]] form {{math|(''x''(''t''),''y''(''t''))}} in some interval {{math|[''a'',''b'']}}, the volumes of the solids generated by revolving the curve around the {{mvar|x}}-axis or the {{mvar|y}}-axis are given by<ref>{{cite book
|title=Application Of Integral Calculus
|first=A.&nbsp;K.
|last=Sharma
|publisher=Discovery Publishing House
|year=2005
|isbn=81-7141-967-4
|page=168
|url=https://books.google.com/books?id=V_WxjYMKuUAC&pg=PA168}}</ref>
<math display="block">\begin{align}
V_x &= \int_a^b \pi y^2 \, \frac{dx}{dt} \, dt \, , \\
V_y &= \int_a^b \pi x^2 \, \frac{dy}{dt} \, dt \, .
\end{align}</math>

Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the {{mvar|x}}-axis or the {{mvar|y}}-axis are given by<ref>{{cite book
|title=Engineering Mathematics
|edition=6th
|first=Ravish R.
|last=Singh
|publisher=Tata McGraw-Hill
|year=1993
|isbn=0-07-014615-2
|page=6.90
|url=https://books.google.com/books?id=oQ1y1HCpeowC&pg=SA6-PA90}}</ref>
<math display="block">\begin{align}
A_x &= \int_a^b 2 \pi y \, \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \, , \\
A_y &= \int_a^b 2 \pi x \, \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \, .
\end{align}</math>

This can also be derived from multivariable integration. If a plane curve is given by <math>\langle x(t), y(t) \rangle</math> then its corresponding surface of revolution when revolved around the x-axis has Cartesian coordinates given by <math>\mathbf{r}(t, \theta) = \langle y(t)\cos(\theta), y(t)\sin(\theta), x(t)\rangle</math> with <math>0 \leq \theta \leq 2\pi</math>. Then the surface area is given by the [[surface integral]]  
<math display="block">A_x = \iint_S dS = \iint_{[a, b] \times [0, 2\pi]} \left\|\frac{\partial \mathbf{r}}{\partial t} \times \frac{\partial \mathbf{r}}{\partial \theta}\right\|\ d\theta\ dt = \int_a^b \int_0^{2\pi} \left\|\frac{\partial \mathbf{r}}{\partial t} \times \frac{\partial \mathbf{r}}{\partial \theta}\right\|\ d\theta\ dt.</math> 

Computing the partial derivatives yields
<math display="block">\frac{\partial \mathbf{r}}{\partial t} = \left\langle \frac{dy}{dt} \cos(\theta), \frac{dy}{dt} \sin(\theta), \frac{dx}{dt} \right\rangle,</math>
<math display="block">\frac{\partial \mathbf{r}}{\partial \theta} = \left\langle -y \sin(\theta), y \cos(\theta), 0 \right\rangle</math>
and computing the [[cross product]] yields 
<math display="block">\frac{\partial \mathbf{r}}{\partial t} \times \frac{\partial \mathbf{r}}{\partial \theta} = \left\langle y \cos(\theta)\frac{dx}{dt}, y \sin(\theta)\frac{dx}{dt}, y \frac{dy}{dt} \right\rangle = y \left\langle \cos(\theta)\frac{dx}{dt}, \sin(\theta)\frac{dx}{dt}, \frac{dy}{dt} \right\rangle </math>
where the trigonometric identity <math>\sin^2(\theta) + \cos^2(\theta) = 1</math> was used. With this cross product, we get
<math display="block">\begin{align}
A_x
&= \int_a^b \int_0^{2\pi} \left\|\frac{\partial \mathbf{r}}{\partial t} \times \frac{\partial \mathbf{r}}{\partial \theta}\right\|\ d\theta\ dt \\[1ex]
&= \int_a^b \int_0^{2\pi} \left\|y \left\langle y \cos(\theta)\frac{dx}{dt}, y \sin(\theta)\frac{dx}{dt}, y \frac{dy}{dt} \right\rangle\right\|\ d\theta\ dt \\[1ex]
&= \int_a^b \int_0^{2\pi} y \sqrt{\cos^2(\theta)\left(\frac{dx}{dt} \right)^2 + \sin^2(\theta)\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\ d\theta\ dt \\[1ex]
&= \int_a^b \int_0^{2\pi} y \sqrt{\left(\frac{dx}{dt} \right)^2 + \left(\frac{dy}{dt} \right)^2}\ d\theta\ dt \\[1ex]
&= \int_a^b 2\pi y \sqrt{\left(\frac{dx}{dt} \right)^2 + \left(\frac{dy}{dt} \right)^2}\ dt
\end{align}</math>
where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar.

== Polar form ==
For a polar curve <math>r=f(\theta)</math> where <math>\alpha\leq \theta\leq \beta</math> and <math>f(\theta) \geq 0</math>, the volumes of the solids generated by revolving the curve around the x-axis or y-axis are
<math display="block">\begin{align}
V_x &= \int_\alpha^\beta \left(\pi r^2\sin^2{\theta} \cos{\theta}\, \frac{dr}{d\theta}-\pi r^3\sin^3{\theta}\right)d\theta\,, \\
V_y &= \int_\alpha^\beta \left(\pi r^2\sin{\theta} \cos^2{\theta}\, \frac{dr}{d\theta}+\pi r^3\cos^3{\theta}\right)d\theta \, .
\end{align}</math>

The areas of the surfaces of the solids generated by revolving the curve around the {{mvar|x}}-axis or the {{mvar|y}}-axis are given
<math display="block">\begin{align}
A_x &= \int_\alpha^\beta 2 \pi r\sin{\theta} \, \sqrt{ r^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta \, , \\
A_y &= \int_\alpha^\beta 2 \pi r\cos{\theta} \, \sqrt{ r^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta \, ,
\end{align}</math>

==See also==
{{Commons category|Solids of revolution}}
* [[Cylindrical symmetry]]
* [[Gabriel's Horn]]
* [[Guldinus theorem]]
* [[Pseudosphere]]
* [[Surface of revolution]]
* [[Ungula]]

==Notes==
{{Reflist}}

== References ==
*{{cite web |website=CliffsNotes.com |title=Volumes of Solids of Revolution |date=12 Apr 2011 |url=http://www.cliffsnotes.com/study_guide/topicArticleId-39909,articleId-39907.html |url-status=dead |archive-url=https://web.archive.org/web/20120319195953/http://www.cliffsnotes.com/study_guide/topicArticleId-39909,articleId-39907.html |archive-date=2012-03-19 }} 
*{{cite book|author1-link=Frank J. Ayres |first1=Frank |last1=Ayres |author2-link=Elliott Mendelson |first2=Elliott |last2=Mendelson |series=[[Schaum's Outlines]] |title=Calculus |publisher=McGraw-Hill Professional |date=2008 |ISBN=978-0-07-150861-2 |pages=244–248}} ({{Google books|Ag26M8TII6oC|online copy|page=244}})
*{{MathWorld |id=SolidofRevolution |title=Solid of Revolution}}

{{Authority control}}

[[Category:Integral calculus]]
[[Category:Solids]]