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en>FrescoBot: Bot: link syntax and minor changes

2014-12-27T00:53:09Z

Bot: link syntax and minor changes

en>Monkbot: /* Definition */Fix CS1 deprecated date parameter errors

2014-02-04T09:44:06Z

Definition: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 11:44, 4 February 2014
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	~~In [[mathematics]]—~~in ~~particular~~, in [~~[multivariable calculus]]—a '''volume integral''' refers to an [[integral]] over a 3-[[dimension]]al domain~~.

	~~It can also mean a [[multiple integral\|triple integral~~]~~] within a region ''D''~~ in ~~'''R'''~~<~~sup~~>3<~~/sup~~> of ~~a [[function (mathematics)\|function]] <math>f(x,y,z)~~,<~~/math~~> ~~and is usually written as:~~

	:<~~math~~>~~\iiint\limits_D f(x~~,~~y,z)\,dx\,dy\,dz~~.~~</math>~~

	~~A volume integral in [~~[~~cylindrical coordinates]] is~~

	:~~<math>\iiint\limits_D f(r,\theta,z)\,r\,dr\,d\theta\,dz, <~~/~~math>~~

	and a ~~volume integral in [[spherical coordinates]] (using the convention for angles with~~ <~~math~~>~~\theta~~<~~/math~~> ~~as the azimuth~~ and ~~<math>\phi</math> measured from the polar axis (see more on [[Spherical coordinate system#Conventions\|conventions]]~~)~~) has~~ the ~~form~~

	:<~~math~~>~~\iiint\limits_D f(\rho,\theta,\phi)\,\rho^2 \sin\phi \,d\rho \,d\theta\, d\phi .~~<~~/math~~>

	~~== Example 1 ==~~

	~~Integrating~~ the ~~function <math> f(x~~,~~y,z) = 1 </math>~~ over ~~a unit cube yields~~ the ~~following result~~:

	~~<math> \int\limits_0^1\int\limits_0^1\int\limits_0^1 1 \,dx\, dy \,dz = \int\limits_0^1\int\limits_0^1 (1 - 0) \,dy \,dz = \int\limits_0^1 (1 - 0) dz = 1 - 0 = 1<~~/~~math>~~

	~~So the volume~~ of the ~~unit cube~~ is ~~1 as expected~~. This ~~is rather trivial however~~, ~~and~~ a ~~volume integral is far more powerful~~. ~~For instance if we have a scalar function~~ <~~math~~>~~\begin{align} f\colon \mathbb{R}^3 &\to \mathbb{R} \end{align}~~<~~/math~~> ~~describing~~ the ~~density of the cube at a given point <math> (x~~,y,z) <~~/math~~> by <~~math~~> ~~f = x+y+z </math> then performing~~ the ~~volume integral will give~~ the ~~total mass~~ of the ~~cube:~~

	~~<math> \int\limits_0^1\int\limits_0^1\int\limits_0^1 \left~~(~~x + y + z\right) \~~, ~~dx \,dy \,dz = \int\limits_0^1\int\limits_0^1 \left(\frac 12 + y + z\right~~) \, ~~dy \,dz = \int \limits_0^1 \left(1 + z\right) \, dz = \frac 32</math>~~

	~~==See also==~~
	*[[Divergence theorem]]
	*[[Surface integral]]
	*[[Volume element]]

	~~==External links==~~
	* {{springer\|title=Multiple integral\|id=p/m065370}}
	* {{MathWorld\|VolumeIntegral\|Volume integral}}

	~~[[Category:Multivariable calculus]]~~

	~~{{mathanalysis-stub}}~~

en>Atlesn: Replace the term 'concious countersterring' with the term 'deliberate countersterring' throughout the article to improve consistency.

2013-12-30T21:32:45Z

Replace the term 'concious countersterring' with the term 'deliberate countersterring' throughout the article to improve consistency.

← Older revision		Revision as of 23:32, 30 December 2013
Line 1:		Line 1:
	~~I would like to introduce myself to you~~, ~~I am Hosea and I entirely dig that name. My spouse and I selected to reside~~ in ~~New Jersey. One of the factors I enjoy most is badge collecting and I am going~~ to ~~be beginning~~ a ~~thing else alongside with it~~. ~~My day job is~~ a ~~library assistant but before long I will be on my very own. I have been functioning on my~~ [~~http:~~//~~Www~~.~~Encyclopedia.com~~/~~searchresults.aspx?q=website website~~] for ~~some time now. Check out it out listed here: http:~~//~~www~~.~~wmhelp.cz~~/~~html/modules.php?name~~=~~Your_Account&op~~=~~userinfo&username~~=~~JRoa~~<br><br>~~Here~~ is ~~my web-site~~ ..~~. [http:~~//~~www.wmhelp.cz~~/~~html~~/~~modules.php?name~~=~~Your_Account&op~~=~~userinfo&username~~=~~JRoa Asics Gel Cumulus 15 Hombre~~]		{{Calculus \|Multivariable}}

			In [[mathematics]]—in particular, in [[multivariable calculus]]—a '''volume integral''' refers to an [[integral]] over a 3-[[dimension]]al domain.

			It can also mean a [[multiple integral\|triple integral]] within a region ''D'' in '''R'''<sup>3</sup> of a [[function (mathematics)\|function]] <math>f(x,y,z),</math> and is usually written as:

			:<math>\iiint\limits_D f(x,y,z)\,dx\,dy\,dz.</math>

			A volume integral in [[cylindrical coordinates]] is

			:<math>\iiint\limits_D f(r,\theta,z)\,r\,dr\,d\theta\,dz, </math>

			and a volume integral in [[spherical coordinates]] (using the convention for angles with <math>\theta</math> as the azimuth and <math>\phi</math> measured from the polar axis (see more on [[Spherical coordinate system#Conventions\|conventions]])) has the form

			:<math>\iiint\limits_D f(\rho,\theta,\phi)\,\rho^2 \sin\phi \,d\rho \,d\theta\, d\phi .</math>

			== Example 1 ==

			Integrating the function <math> f(x,y,z) = 1 </math> over a unit cube yields the following result:

			<math> \int\limits_0^1\int\limits_0^1\int\limits_0^1 1 \,dx\, dy \,dz = \int\limits_0^1\int\limits_0^1 (1 - 0) \,dy \,dz = \int\limits_0^1 (1 - 0) dz = 1 - 0 = 1</math>

			So the volume of the unit cube is 1 as expected. This is rather trivial however, and a volume integral is far more powerful. For instance if we have a scalar function <math>\begin{align} f\colon \mathbb{R}^3 &\to \mathbb{R} \end{align}</math> describing the density of the cube at a given point <math> (x,y,z) </math> by <math> f = x+y+z </math> then performing the volume integral will give the total mass of the cube:

			<math> \int\limits_0^1\int\limits_0^1\int\limits_0^1 \left(x + y + z\right) \, dx \,dy \,dz = \int\limits_0^1\int\limits_0^1 \left(\frac 12 + y + z\right) \, dy \,dz = \int \limits_0^1 \left(1 + z\right) \, dz = \frac 32</math>

			==See also==
			*[[Divergence theorem]]
			*[[Surface integral]]
			*[[Volume element]]

			==External links==
			* {{springer\|title=Multiple integral\|id=p/m065370}}
			* {{MathWorld\|VolumeIntegral\|Volume integral}}

			[[Category:Multivariable calculus]]

			{{mathanalysis-stub}}

en>AndrewDressel: Break out section on the definition and add reference with quotation.

2012-08-01T14:55:05Z

Break out section on the definition and add reference with quotation.

New page

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