Solid of revolution

Two common methods for finding the volume of a solid of revolution are the disc method and the 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.

File:Disc integration.svg

{{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

$$V\; =\; \backslash pi\; \backslash int\_a^b\; \backslash left|\; f(y)^2\; -\; g(y)^2\backslash right|\backslash ,dy\backslash ,\; .$$

If {{math|1=g(y) = 0}} (e.g. revolving an area between the curve and the {{mvar|y}}-axis), this reduces to:

$$V\; =\; \backslash pi\; \backslash int\_a^b\; f(y)^2\; \backslash ,dy\backslash ,\; .$$

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(R² − r²)}}, 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|πf(y)² 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):

$$V\; =\; \backslash iiint\_D\; dV\; =\; \backslash int\_a^b\; \backslash int\_\{g(z)\}^\{f(z)\}\; \backslash int\_0^\{2\backslash pi\}\; r\backslash ,d\backslash theta\backslash ,dr\backslash ,dz\; =\; 2\backslash pi\; \backslash int\_a^b\backslash int\_\{g(z)\}^\{f(z)\}\; r\backslash ,dr\backslash ,dz\; =\; 2\backslash pi\; \backslash int\_a^b\; \backslash frac\{1\}\{2\}r^2\backslash Vert^\{f(z)\}\_\{g(z)\}\; \backslash ,dz\; =\; \backslash pi\; \backslash int\_a^b\; (f(z)^2\; -\; g(z)^2)\backslash ,dz$$

{{main|Shell integration}}

File:Shell integration.svg

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

$$V\; =\; 2\backslash pi\; \backslash int\_a^b\; x\; |f(x)\; -\; g(x)|\backslash ,\; dx\backslash ,\; .$$

If {{math|1=g(x) = 0}} (e.g. revolving an area between curve and {{mvar|x}}-axis), this reduces to:

$$V\; =\; 2\backslash pi\; \backslash int\_a^b\; x\; |\; f(x)\; |\; \backslash ,dx\backslash ,\; .$$

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|2πrh}}, 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:

$$V\; =\; \backslash iiint\_D\; dV\; =\; \backslash int\_a^b\; \backslash int\_\{g(r)\}^\{f(r)\}\; \backslash int\_0^\{2\backslash pi\}\; r\backslash ,d\backslash theta\backslash ,dz\backslash ,dr\; =\; 2\backslash pi\; \backslash int\_a^b\backslash int\_\{g(r)\}^\{f(r)\}\; r\backslash ,dz\backslash ,dr\; =\; 2\backslash pi\backslash int\_a^b\; r(f(r)\; -\; g(r))\backslash ,dr.$$

{{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

}}

File:Paolo uccello, studio di vaso in prospettiva 02.jpg: study of a vase as a solid of revolution by Paolo Uccello. 15th century]]

When a curve is defined by its 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{{cite book

|title=Application Of Integral Calculus

|first=A. 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}}

$$\backslash begin\{align\}$$

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}

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{{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}}

$$\backslash begin\{align\}$$

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}

This can also be derived from multivariable integration. If a plane curve is given by $\backslash langle\; x(t),\; y(t)\; \backslash rangle$ then its corresponding surface of revolution when revolved around the x-axis has Cartesian coordinates given by $\backslash mathbf\{r\}(t,\; \backslash theta)\; =\; \backslash langle\; y(t)\backslash cos(\backslash theta),\; y(t)\backslash sin(\backslash theta),\; x(t)\backslash rangle$ with $0\; \backslash leq\; \backslash theta\; \backslash leq\; 2\backslash pi$. Then the surface area is given by the surface integral

$$A\_x\; =\; \backslash iint\_S\; dS\; =\; \backslash iint\_\{[a,\; b]\; \backslash times\; [0,\; 2\backslash pi]\}\; \backslash left\backslash |\backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; t\}\; \backslash times\; \backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; \backslash theta\}\backslash right\backslash |\backslash \; d\backslash theta\backslash \; dt\; =\; \backslash int\_a^b\; \backslash int\_0^\{2\backslash pi\}\; \backslash left\backslash |\backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; t\}\; \backslash times\; \backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; \backslash theta\}\backslash right\backslash |\backslash \; d\backslash theta\backslash \; dt.$$

Computing the partial derivatives yields

$$\backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; t\}\; =\; \backslash left\backslash langle\; \backslash frac\{dy\}\{dt\}\; \backslash cos(\backslash theta),\; \backslash frac\{dy\}\{dt\}\; \backslash sin(\backslash theta),\; \backslash frac\{dx\}\{dt\}\; \backslash right\backslash rangle,$$

$$\backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; \backslash theta\}\; =\; \backslash left\backslash langle\; -y\; \backslash sin(\backslash theta),\; y\; \backslash cos(\backslash theta),\; 0\; \backslash right\backslash rangle$$

and computing the cross product yields

$$\backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; t\}\; \backslash times\; \backslash frac\{\backslash partial\; \backslash mathbf\{r\}\}\{\backslash partial\; \backslash theta\}\; =\; \backslash left\backslash langle\; y\; \backslash cos(\backslash theta)\backslash frac\{dx\}\{dt\},\; y\; \backslash sin(\backslash theta)\backslash frac\{dx\}\{dt\},\; y\; \backslash frac\{dy\}\{dt\}\; \backslash right\backslash rangle\; =\; y\; \backslash left\backslash langle\; \backslash cos(\backslash theta)\backslash frac\{dx\}\{dt\},\; \backslash sin(\backslash theta)\backslash frac\{dx\}\{dt\},\; \backslash frac\{dy\}\{dt\}\; \backslash right\backslash rangle$$

where the trigonometric identity $\backslash sin^2(\backslash theta)\; +\; \backslash cos^2(\backslash theta)\; =\; 1$ was used. With this cross product, we get

$$\backslash begin\{align\}$$

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}

where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar.

For a polar curve $r=f(\backslash theta)$ where $\backslash alpha\backslash leq\; \backslash theta\backslash leq\; \backslash beta$ and $f(\backslash theta)\; \backslash geq\; 0$, the volumes of the solids generated by revolving the curve around the x-axis or y-axis are

$$\backslash begin\{align\}$$

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}

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

$$\backslash begin\{align\}$$

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}

{{Commons category|Solids of revolution}}

• Cylindrical symmetry
• Gabriel's Horn
• Guldinus theorem
• Pseudosphere
• Surface of revolution
• Ungula

{{Reflist}}

• {{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