BREAKING NEWS

## Summary

In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities.

## In coordinates

It can also mean a triple integral within a region $D\subset \mathbb {R} ^{3}$  of a function $f(x,y,z),$  and is usually written as:

$\iiint _{D}f(x,y,z)\,dx\,dy\,dz.$

A volume integral in cylindrical coordinates is

$\iiint _{D}f(\rho ,\varphi ,z)\rho \,d\rho \,d\varphi \,dz,$

and a volume integral in spherical coordinates (using the ISO convention for angles with $\varphi$  as the azimuth and $\theta$  measured from the polar axis (see more on conventions)) has the form
$\iiint _{D}f(r,\theta ,\varphi )r^{2}\sin \theta \,dr\,d\theta \,d\varphi .$

## Example

Integrating the equation $f(x,y,z)=1$  over a unit cube yields the following result:

$\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}1\,dx\,dy\,dz=\int _{0}^{1}\int _{0}^{1}(1-0)\,dy\,dz=\int _{0}^{1}\left(1-0\right)dz=1-0=1$

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 density function on the unit cube then the volume integral will give the total mass of the cube. For example for density function:

${\begin{cases}f:\mathbb {R} ^{3}\to \mathbb {R} \\f:(x,y,z)\mapsto x+y+z\end{cases}}$

the total mass of the cube is:
$\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}(x+y+z)\,dx\,dy\,dz=\int _{0}^{1}\int _{0}^{1}\left({\frac {1}{2}}+y+z\right)dy\,dz=\int _{0}^{1}(1+z)\,dz={\frac {3}{2}}$
{"ft_lang_label":"__label__en","ft_lang_prob":0.8766955,"math_prob":0.999984,"size":797,"snap":"2022-05-2022-21","text_gpt3_token_len":160,"char_repetition_ratio":0.1626734,"word_repetition_ratio":0.0,"special_character_ratio":0.19196989,"punctuation_ratio":0.0952381,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999813,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-20T01:07:51Z\",\"WARC-Record-ID\":\"<urn:uuid:808661e8-bd19-4fc2-852c-78735285d127>\",\"Content-Length\":\"93219\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5a871c2f-937f-400c-9099-cacf7832487b>\",\"WARC-Concurrent-To\":\"<urn:uuid:67106a61-4385-429a-aa5e-57fb63f97014>\",\"WARC-IP-Address\":\"18.224.181.193\",\"WARC-Target-URI\":\"https://www.knowpia.com/knowpedia/Volume_integral\",\"WARC-Payload-Digest\":\"sha1:2RCPQHONF76YZGQKW3DM4H7C2Y6AUYN6\",\"WARC-Block-Digest\":\"sha1:UZTFPCDK4QIBX536N6WKAUQQ2ND2OPRC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662530553.34_warc_CC-MAIN-20220519235259-20220520025259-00408.warc.gz\"}"}