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Area Mathematics - General concepts and linear algebra / Geometry

IEV ref102-04-33

en
area
unique positive value, if it exists, associated with a subset of a surface in the three-dimensional Euclidean space, with the following properties:

  • for a rectangle, the value is the product of the two side lengths,
  • for a disjoint union of subsets, the value is the sum of the values associated with these subsets,
  • for more complicated subsets, the value can be approximated by sums and given by an integral

Note 1 to entry: For the portion of plane limited by the straight lines x = a, x = b, y = 0 and the arc of curve y = f(x) with a < b and f(x) ≥ 0, the area is a b f(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapehabaGaamOzai aacIcacaWG4bGaaiykaiGacsgacaWG4baaleaacaWGHbaabaGaamOy aaqdcqGHRiI8aaaa@3EAD@ .

Note 2 to entry: For a surface defined by r=f(u,v) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahkhacqGH9aqpca WHMbGaaiikaiaadwhacaaMc8UaaiilaiaaykW7caWG2bGaaiykaaaa @42D1@ where (u,v)U R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacIcacaWG1bGaai ilaiaadAhacaGGPaGaeyicI4CcLbuacaqGvbGccqGHgksZieqacaWF sbWaaWbaaSqabeaajugabiaabkdaaaaaaa@440F@ , the area is U | f u f u |dudv MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapifabaWaaqWaae aadaWcaaqaaiabgkGi2kaahAgaaeaacqGHciITcaWG1baaaiabgwSi xpaalaaabaGaeyOaIyRaaCOzaaqaaiabgkGi2kaadwhaaaaacaGLhW UaayjcSdGaeyyXICTaciizaiaadwhaciGGKbGaamODaaWcbaGaaeyv aaqab0Gaey4kIiVaey4kIipaaaa@4F47@ .

Note 3 to entry: For a surface defined by the equation z = f(x, y), the area is S 1+ ( f x ) 2 + ( f y ) 2 dxdy MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapifabaWaaOaaae aacaaIXaGaey4kaSYaaeWaaeaadaWcaaqaaiabgkGi2kaadAgaaeaa cqGHciITcaWG4baaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaa aakiabgUcaRmaabmaabaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOa IyRaamyEaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaabe aakiGacsgacaWG4bGaciizaiaadMhaaSqaaiaabofaaeqaniabgUIi YlabgUIiYdaaaa@4F1B@ .

Note 4 to entry: In the usual geometrical space, the area of a surface is a quantity of the dimension length squared.


fr
aire, f
valeur positive unique, si elle existe, associée à un sous-ensemble d'une surface dans l'espace euclidien à trois dimensions, avec les propriétés suivantes:

  • pour un rectangle, la valeur est le produit des longueurs des côtés,
  • pour une union disjointe de sous-ensembles, la valeur est la somme des valeurs qui leur sont associées,
  • pour des sous-ensembles plus compliqués, la valeur peut être approchée par des sommes et donnée par une intégrale

Note 1 à l'article: Pour la partie d'un plan limitée par les droites x = a, x = b, y = 0 et l'arc de courbe y = f(x) avec a < b et f(x) ≥ 0, l'aire est a b f(x)dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapehabaGaamOzai aacIcacaWG4bGaaiykaiGacsgacaWG4baaleaacaWGHbaabaGaamOy aaqdcqGHRiI8aaaa@3EAD@ .

Note 2 à l'article: Pour une surface définie par r=f(u,v) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaahkhacqGH9aqpca WHMbGaaiikaiaadwhacaaMc8UaaiilaiaaykW7caWG2bGaaiykaaaa @42D1@ , où (u,v)U R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqk0di9Wr=fpeei0di9v8qiW7rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaaiaacIcacaWG1bGaai ilaiaadAhacaGGPaGaeyicI4CcLbuacaqGvbGccqGHgksZieqacaWF sbWaaWbaaSqabeaajugabiaabkdaaaaaaa@440F@ , l'aire est U | f u f u |dudv MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapifabaWaaqWaae aadaWcaaqaaiabgkGi2kaahAgaaeaacqGHciITcaWG1baaaiabgwSi xpaalaaabaGaeyOaIyRaaCOzaaqaaiabgkGi2kaadwhaaaaacaGLhW UaayjcSdGaeyyXICTaciizaiaadwhaciGGKbGaamODaaWcbaGaaeyv aaqab0Gaey4kIiVaey4kIipaaaa@4F47@ .

Note 3 à l'article: Pour une surface définie par l'équation z = f(x, y), l'aire est S 1+ ( f x ) 2 + ( f y ) 2 dxdy MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaeaaciWacmGadaGadeaabaGaaqaaaOqaamaapifabaWaaOaaae aacaaIXaGaey4kaSYaaeWaaeaadaWcaaqaaiabgkGi2kaadAgaaeaa cqGHciITcaWG4baaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaa aakiabgUcaRmaabmaabaWaaSaaaeaacqGHciITcaWGMbaabaGaeyOa IyRaamyEaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaabe aakiGacsgacaWG4bGaciizaiaadMhaaSqaaiaabofaaeqaniabgUIi YlabgUIiYdaaaa@4F1B@ .

Note 4 à l'article: Dans l'espace géométrique usuel, l'aire d'une surface est une grandeur ayant la dimension du carré d'une longueur.


de
Flächeninhalt, m

es
área

ja
面積

pl
pole powierzchni
pole (1)

pt
área

sr
површина, ж јд

sv
area

zh
面积

Publication date: 2008-08
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