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[[File:Surface of revolution illustration.png|thumb|A portion of the curve ''x''=2+cos ''z'' rotated around the ''z'' axis]]
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A '''surface of revolution''' is a [[surface]] in [[Euclidean space]] created by rotating a [[curve]] (the '''generatrix''') around a [[straight line]] in its plane (the '''axis''').<ref>''Analytic Geometry'' Middlemiss, Marks, and Smart. 3rd Edition Ch. 15 Surfaces and Curves, &sect; 15-4 Surfaces of Revolution {{LCCN|68015472}} pp 378 ff.</ref>
 
Examples of surfaces generated by a straight line are [[cylinder (geometry)|cylindrical]] and [[conical surface]]s when the line is coplanar with the axis, as well as [[Hyperboloid|hyperboloids of one sheet]] when the line is [[Skew lines|skew]] to the axis. A circle that is rotated about its center point generates a sphere, and if the circle is rotated about a coplanar axis, not crossing the circle, then it generates a [[torus]].
 
==Area formula==
If the curve is described by the [[parametric curve|parametric]] functions <math>x(t)</math>, <math>y(t)</math>, with <math>t</math> ranging over some interval <math>[a,b]</math>, and the axis of revolution is the <math>y</math>-axis, then the area <math>A_y</math> is given by the [[integral]]
 
:<math> A_y = 2 \pi \int_a^b x(t) \ \sqrt{\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2} \, dt, </math>
 
provided that <math>x(t)</math> is never negative between the endpoints a and b. This formula is the calculus equivalent of  [[Pappus's centroid theorem]].<ref>''Calculus'', George B. Thomas, 3rd Edition, Ch. 6 Applications of the definite integral, &sect;&sect; 6.7,6.11, Area of a Surface of Revolution pp 206-209, The Theorems of Pappus, pp 217-219 {{LCCN|69016407}}</ref> The quantity
 
:<math>\sqrt{ \left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2 }</math>
 
comes from the [[Pythagorean theorem]] and represents a small segment of the arc of the curve, as in the [[arc length]] formula. The quantity <math>2\pi x(t)</math> is the path of (the centroid of) this small segment, as required by Pappus' theorem.
 
Likewise, when the axis of rotation is the <math>x</math>-axis and provided that <math>y(t)</math> is never negative, the area is given by<ref>{{cite book
|title=Engineering Mathematics
|edition=6
|author=Singh
|publisher=Tata McGraw-Hill
|year=1993
|isbn=0-07-014615-2
|page=6.90
|url=http://books.google.com/books?id=oQ1y1HCpeowC}}, [http://books.google.com/books?id=oQ1y1HCpeowC&pg=SA6-PA90 Chapter 6, page 6.90]
</ref>
:<math> A_x = 2 \pi \int_a^b y(t) \ \sqrt{\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2} \, dt. </math>
 
If the curve is described by the function ''y'' = ''f(x)'', ''a'' ≤ ''x'' ≤ ''b'', then the integral becomes
 
:<math>A_x = 2\pi\int_a^b y \sqrt{1+\left(\frac{dy}{dx}\right)^2} \, dx = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx</math>
 
for revolution around the ''x''-axis, and
 
:<math>A_y =2\pi\int_a^b x \sqrt{1+\left(\frac{dx}{dy}\right)^2} \, dy</math>
 
for revolution around the ''y''-axis (Using ''a'' ≤ ''y'' ≤ ''b''). These come from the above formula.
 
For example, the [[sphere|spherical surface]] with unit radius is generated by the curve ''x''(''t'') = sin(''t''), ''y''(''t'') = cos(''t''), when ''t'' ranges over <math>[0,\pi]</math>. Its area is therefore
:<math>\begin{align}
A
&{}= 2 \pi \int_0^\pi \sin(t) \sqrt{\left(\cos(t)\right)^2 + \left(\sin(t)\right)^2} \, dt \\
&{}= 2 \pi \int_0^\pi \sin(t) \, dt \\
&{}= 4\pi.
\end{align}</math>
 
For the case of the spherical curve with radius <math>r \,</math>, <math>y(x) = \sqrt{r^2 - x^2}</math> rotated about the ''x''-axis
:<math>\begin{align}
A
&{}= 2 \pi \int_{-r}^{r} \sqrt{r^2 - x^2}\,\sqrt{1 + \frac{x^2}{r^2 - x^2}}\,dx \\
&{}= 2 \pi r\int_{-r}^{r} \,\sqrt{r^2 - x^2}\,\sqrt{\frac{1}{r^2 - x^2}}\,dx \\
&{}= 2 \pi r\int_{-r}^{r} \,dx \\
&{}= 4 \pi r^2\,
\end{align}</math>
 
A [[minimal surface of revolution]] is the surface of revolution of the curve between two given points which [[mathematical optimization|minimizes]] [[surface area]].<ref name="Mathworld: Minimal Surface of Revolution">{{cite web | url=http://mathworld.wolfram.com/MinimalSurfaceofRevolution.html | title=Minimal Surface of Revolution | last=Weisstein | first=Eric W. | authorlink=Eric W. Weisstein | work=[[Mathworld]] | publisher=[[Wolfram Research]] | accessdate=2012-08-29}}</ref> A basic problem in the [[calculus of variations]] is finding the curve between two points that produces this minimal surface of revolution.<ref name="Mathworld: Minimal Surface of Revolution"/>
 
==Rotating a function==
To generate a surface of revolution out of any 2-dimensional scalar function <math>y=f(x)</math>, simply make <math>u</math> the function's parameter, set the axis of rotation's function to simply <math>u</math>, then use <math>v</math> to rotate the function around the axis by setting the other two functions equal to <math>f(u)\sin v </math> and <math>f(u)\cos v</math>. For example, to rotate a function <math>y=f(x)</math> around the x-axis starting from the top of the <math>xz</math>-plane, parameterize it as <math>\vec r(u,v)=\langle u,f(u)\sin v,f(u)\cos v\rangle</math> for <math>u=x</math> and <math>v\in[0,2\pi]</math> .
 
==Geodesics on a surface of revolution==
Geodesics on a surface of revolution are governed by [[Clairaut's relation]].
 
==Applications of surfaces of revolution==
The use of surfaces of revolution is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to determine surface area without the use of measuring the length and radius of the object being designed.
 
==See also==
* [[Channel surface]], a generalisation of a surface of revolution
* [[Gabriel's Horn]]
* [[Liouville surface]], another generalization of a surface of revolution
* [[Solid of revolution]]
* [[Surface integral]]
 
==References==
<references/>
 
==External links==
*{{MathWorld|title=Surface of Revolution|urlname=SurfaceofRevolution}}
*[http://www.mathcurve.com/surfaces/revolution/revolution.shtml "Surface de révolution" at Encyclopédie des Formes Mathématiques Remarquables]
 
{{DEFAULTSORT:Surface Of Revolution}}
[[Category:Integral calculus]]
[[Category:Surfaces]]

Latest revision as of 21:42, 7 January 2015

We usually find convenient methods to accelerate computer by making the many out of the built inside tools inside a Windows and also downloading the Service Pack updates-speed up your PC plus fix error. Simply follow a limited protocols to quickly create the computer rapidly than ever.

Google Chrome crashes on Windows 7 by the corrupted cache contents and difficulties with all the stored browsing data. Delete the browsing data plus obvious the contents of the cache to resolve this issue.

Your PC might have a fragmented difficult drive or the windows registry could have been corrupted. It may moreover be due to the dust and dirt that should be cleaned. Whatever the issue, you can always find a solution. Here are several tricks on how to create the PC run faster.

First, always clean your PC plus keep it free of dust and dirt. Dirt clogs up all of the fans plus will cause the PC to overheat. We furthermore have to clean up disk space in order to create a computer run quicker. Delete temporary and unnecessary files plus unused programs. Empty the recycle bin and remove programs you're not using.

These are the results that the tuneup utilities 2014 found: 622 wrong registry entries, 45,810 junk files, 15,643 unprotected privacy files, 8,462 bad Active X goods which were not blocked, 16 performance features which were not optimized, plus 4 updates that the computer needed.

Another key element when you compare registry cleaners is having a center to manage a start-up jobs. This just signifies that you can choose what programs you want to start when you start a PC. If you have unnecessary programs beginning when we boot up your PC this will lead to a slow running computer.

The 'registry' is simply the central database that stores all your settings and choices. It's a absolutely significant part of the XP program, meaning which Windows is constantly adding plus updating the files inside it. The issues occur when Windows actually corrupts & loses a few of these files. This makes the computer run slow, because it attempts difficult to locate them again.

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