Countersteering: Difference between revisions

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{{Calculus |Multivariable}}
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In [[mathematics]]&mdash;in particular, in [[multivariable calculus]]&mdash;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}}

Revision as of 11:44, 4 February 2014

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