# Surface of revolution

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Surface of revolution

A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of rotation) that lies on the same plane.

Examples of surfaces generated by a straight line are the cylindrical and conical surfaces. A circle that is rotated about a (coplanar) axis through the center generates a sphere. If the axis is coplanar and outside the circle it generates a toroidal surface.

If the curve is described by the parametric functions $x\left(t\right)$, $y\left(t\right)$, with $t$ ranging over some interval $\left[a,b\right]$, and the axis of revolution is the $y$ axis, then the area $A$ is given by the integral

:$A = 2 pi int_a^b x\left(t\right) sqrt\left\{left\left(\left\{dx over dt\right\} ight\right)^2 + left\left(\left\{dy over dt\right\} ight\right)^2\right\} , dt,$

provided that $x\left(t\right)$ is never negative. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity

:$left\left(\left\{dx over dt\right\} ight\right)^2 + left\left(\left\{dy over dt\right\} ight\right)^2$

comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity $2pi x\left(t\right)$ is the path of (the centroid of) this small segment, as required by Pappus's theorem.

If the curve is described by the function "y" = "f"("x"), "a" &le; "x" &le; "b", then the integral becomes

:$A=2piint_a^b y sqrt\left\{1+left\left(frac\left\{dy\right\}\left\{dx\right\} ight\right)^2\right\} , dx$

for revolution around the "x"-axis, and

:$A=2piint_a^b x sqrt\left\{1+left\left(frac\left\{dx\right\}\left\{dy\right\} ight\right)^2\right\} , dy$

for revolution around the "y"-axis. These come from the above formula.

For example, the spherical surface with unit radius is generated by the curve "x"("t") = sin("t"), "y"("t") = cos("t"), when "t" ranges over $\left[0,pi\right]$. Its area is therefore

:$A = 2 pi int_0^pi sin\left(t\right) sqrt\left\{left\left(cos\left(t\right) ight\right)^2 + left\left(sin\left(t\right) ight\right)^2\right\} , dt = 2 pi int_0^pi sin\left(t\right) , dt = 4pi.$

For the case of the spherical curve with radius "r", $y\left(x\right) = sqrt\left\{r^2 - x^2\right\}$ rotated about the "x"-axis

:$A = 2 pi int_\left\{-r\right\}^\left\{r\right\} sqrt\left\{r^2 - x^2\right\},sqrt\left\{1 + frac\left\{x^2\right\}\left\{r^2 - x^2,dx$:$= 2 pi int_\left\{-r\right\}^\left\{r\right\} r,sqrt\left\{r^2 - x^2\right\},sqrt\left\{frac\left\{1\right\}\left\{r^2 - x^2,dx$:$= 2 pi int_\left\{-r\right\}^\left\{r\right\} r,dx$:$= 4 pi r^2,$

Rotating a function

To generate a surface of revolution out of any 2-dimensional scalar function $y=f\left(x\right)$, simply make $u$ the function's parameter, set the axis of rotation's function to simply $u$, then use $v$ to rotate the function around the axis by setting the other two functions equal to $f\left(u\right)sin v$ and $f\left(u\right)cos v$ conversely. For example, to rotate a function $y=f\left(x\right)$ around the x-axis starting from the top of the $xz$-plane, parameterize it as $vec r\left(u,v\right)=langle u,f\left(u\right)sin v,f\left(u\right)cos v angle$ for $uin x$ and $vin \left[0,2pi\right)$ .

Applications of surfaces of revolution

The use of surface of revolutions 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.

ee also

* Solid of revolution
* Gabriel's Horn
* Channel surface - a generalisation of a surface of revolution
* Liouville surface - another generalization of a surface of revolution

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### Look at other dictionaries:

• Surface de revolution — Surface de révolution Une surface de révolution est une surface paramétrée et orientée de ℝ³, la surface balayée par la rotation d une courbe plane. Les surfaces de révolution comprennent les tores, les sphères, les cylindres, les sphéroïdes, les …   Wikipédia en Français

• Surface de révolution — Une surface de révolution est une surface paramétrée et orientée de ℝ³, la surface balayée par la rotation d une courbe plane, appelée méridienne. Les surfaces de révolution comprennent les tores, les sphères, les cylindres, les sphéroïdes, les… …   Wikipédia en Français

• surface of revolution — a surface held to be formed by the revolution of a plane curve about a line in its plane * * * Math. a surface formed by revolving a plane curve about a given line. [1830 40] * * * surface of revolution, Geometry. a surface which is generated by… …   Useful english dictionary

• surface of revolution — noun A surface formed when a given curve is revolved around a given axis. If the resulting surface is a closed one, it also defines a solid of revolution. A sphere is a surface of revolution of a circle around an axis which runs through the… …   Wiktionary

• surface de révolution — sukimosi paviršius statusas T sritis fizika atitikmenys: angl. surface of revolution; surface of rotation vok. Drehfläche, f; Rotationsfläche, f; Umdrehungsfläche, f rus. поверхность вращения, f pranc. surface de rotation, f; surface de… …   Fizikos terminų žodynas

• surface of revolution — sukimosi paviršius statusas T sritis fizika atitikmenys: angl. surface of revolution; surface of rotation vok. Drehfläche, f; Rotationsfläche, f; Umdrehungsfläche, f rus. поверхность вращения, f pranc. surface de rotation, f; surface de… …   Fizikos terminų žodynas

• surface of revolution — Date: 1840 a surface formed by the revolution of a plane curve about a line in its plane …   New Collegiate Dictionary

• surface of revolution — Math. a surface formed by revolving a plane curve about a given line. [1830 40] * * * …   Universalium

• Surface de révolution d'axe D — ● Surface de révolution d axe D surface engendrée par la rotation autour de D d un arc Γ. (Les sections de cette surface par un plan perpendiculaire à D sont des cercles appelés parallèles.) …   Encyclopédie Universelle

• Vertex of a surface of revolution — Vertex Ver tex, n.; pl. {Vertexes}, L. {Vertices}. [L. vertex, icis, a whirl, top of the head, top, summit, from vertere to turn. See {Verse}, and cf. {Vortex}.] A turning point; the principal or highest point; top; summit; crown; apex.… …   The Collaborative International Dictionary of English