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Texte intégral

(1)

cours 20

3.4 APPLICATIONS ET

AIRE DE SURFACE

(2)

Au dernier cours, nous avons vu

(3)

Au dernier cours, nous avons vu

Coordonnées polaires.

(4)

Au dernier cours, nous avons vu

Coordonnées polaires.

Changement de variables en coordonnées polaires.

(5)

Au dernier cours, nous avons vu

Coordonnées polaires.

Changement de variables en coordonnées polaires.

Une intégrale importante en probabilité.

(6)

Aujourd’hui, nous allons voir

(7)

Aujourd’hui, nous allons voir

Applications de l’intégrale double.

(8)

Aujourd’hui, nous allons voir

Applications de l’intégrale double.

L’aire d’une surface.

(9)

Valeur moyenne

(10)

Valeur moyenne

Aire =

Z b

a

f (x)dx

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(11)

Valeur moyenne

Aire =

Z b

a

f (x)dx

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(12)

Valeur moyenne

Aire =

Z b

a

f (x)dx

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(13)

Valeur moyenne

Aire =

Z b

a

f (x)dx

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= base<latexit sha1_base64="ETtfmXXuDpo/oBaX3IqUiTPJqwU=">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</latexit> ⇥ hauteur

= (b<latexit sha1_base64="A/xBs3SAgu1ztM4eTgUvUHvUP84=">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</latexit> a) ⇥ hauteur

(14)

Valeur moyenne

Aire =

Z b

a

f (x)dx

<latexit sha1_base64="tm98GUGcY/FjB1lcGR83fhMl/5c=">AAADEXicjVLBThRBEH2MKAgqq9z0MnFjgpfNDJjIxQT14hETFkhY3MzM9mKH2ZlJTw/ZzYbEf/AfvOqVm/HqF/AHeOMTfF00RiFGezI9r1/Vq6mqrrTKdW2j6HQmuDF789bc/O2FxTt37y217j/YrsvGZKqblXlpdtOkVrkuVNdqm6vdyqhklOZqJz187ew7R8rUuiy27KRS+6PkoNBDnSWWVL/1sGfV2E5faqOOX/R0YfvJu3S4Mn46GPdb7agTyQqvg9iDNvzaLFvn6GGAEhkajKBQwBLnSFDz2UOMCBW5fUzJGSItdoVjLFDb0EvRIyF7yP2Apz3PFjy7mLWoM/4l52uoDPGEmpJ+htj9LRR7I5Ed+7fYU4npcpvwm/pYI7IW78n+S3fp+b86V4vFEOtSg2ZNlTCuusxHaaQrLvPwt6osI1TkHB7QbogzUV72ORRNLbW73iZiPxNPx7pz5n0b/PBZKhxJ1Mmv7Kdyh5r2Sno5IbLc5ZY4EvHVAbgOtlc78VonfvusvfHKD8c8HuExVjgBz7GBN9hEl9l8wCd8xpfgY3ASfA2+XbgGM16zjD9W8P0nUTal+w==</latexit>

= base<latexit sha1_base64="ETtfmXXuDpo/oBaX3IqUiTPJqwU=">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</latexit> ⇥ hauteur

= (b<latexit sha1_base64="A/xBs3SAgu1ztM4eTgUvUHvUP84=">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</latexit> a) ⇥ hauteur

= (b<latexit sha1_base64="JOQGmTt15uf7gfiFafEyaFeXrvE=">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</latexit> a) ⇥ y¯

(15)

Valeur moyenne

Aire =

Z b

a

f (x)dx

<latexit sha1_base64="tm98GUGcY/FjB1lcGR83fhMl/5c=">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</latexit>

= base<latexit sha1_base64="ETtfmXXuDpo/oBaX3IqUiTPJqwU=">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</latexit> ⇥ hauteur

= (b<latexit sha1_base64="A/xBs3SAgu1ztM4eTgUvUHvUP84=">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</latexit> a) ⇥ hauteur

= (b<latexit sha1_base64="JOQGmTt15uf7gfiFafEyaFeXrvE=">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</latexit> a) ⇥ y¯

(16)

Valeur moyenne

Aire =

Z b

a

f (x)dx

<latexit sha1_base64="tm98GUGcY/FjB1lcGR83fhMl/5c=">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</latexit>

= base<latexit sha1_base64="ETtfmXXuDpo/oBaX3IqUiTPJqwU=">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</latexit> ⇥ hauteur

= (b<latexit sha1_base64="A/xBs3SAgu1ztM4eTgUvUHvUP84=">AAADEnicjVLLThVBED0MPgAfXGHJZuKNCS68mVES3ZgQ3bjExAskXEJ6hgY6zCs9PcSbGxb+g//gFrbuDFt/wD+QnZ/g6bIhKDHak+k5fapOTVV1ZU1hWpck36ai6Rs3b92emZ27c/fe/fneg4X1tu5srod5XdR2M1OtLkylh864Qm82VqsyK/RGdvja2zeOtG1NXb1z40Zvl2q/MnsmV47UTm/p5XL2RD0eOVPqNh45/d5NDlTndGePd3r9ZJDIiq+DNIA+wlqrez8wwi5q5OhQQqOCIy6g0PLZQooEDbltTMhZIiN2jWPMUdvRS9NDkT3kvs/TVmArnn3MVtQ5/1LwtVTGeERNTT9L7P8Wi72TyJ79W+yJxPS5jfnNQqySrMMB2X/pLjz/V+drcdjDC6nBsKZGGF9dHqJ00hWfeXylKscIDTmPd2m3xLkoL/oci6aV2n1vldi/i6dn/TkPvh3OQ5YaRxJ1fJn9RO7Q0N5IL8dEjrvcEkci/XMAroP1p4P02SB9u9JffRWGYwZLeIhlTsBzrOIN1jBkNh/wCSc4jT5Gn6Mv0dkv12gqaBbx24q+/gTYjKYv</latexit> a) ⇥ hauteur

= (b<latexit sha1_base64="JOQGmTt15uf7gfiFafEyaFeXrvE=">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</latexit> a) ⇥ y¯

¯

y = 1 b a

Z b

a

f (x)dx

<latexit sha1_base64="HrIofoNtGlM0/G6k7TCqasuvrKc=">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</latexit>

(17)

Volume =

ZZ

R

f (x, y)dV

<latexit sha1_base64="R2HAeliY84shi3X+5GnIGBJJLsE=">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</latexit>

(18)

Volume =

ZZ

R

f (x, y)dV

<latexit sha1_base64="R2HAeliY84shi3X+5GnIGBJJLsE=">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</latexit>

= Aire(R)<latexit sha1_base64="Ya8HQ8izQ3EYdWbIXmvM43RVsx8=">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</latexit> ⇥ z¯

(19)

Volume =

ZZ

R

f (x, y)dV

<latexit sha1_base64="R2HAeliY84shi3X+5GnIGBJJLsE=">AAADFXicjVJNTxRBEH0MfiB+rXgkIRM3JpiYzQya6MWE4MUjEnchYclmZrYXO8xXZnoIkw03/gP/gStcuRGvnP0HevMn+LpsjEqM9mR6Xr+qV1NVXXGZ6toEwecZb/bGzVu35+7M3713/8HDzqOFQV00VaL6SZEW1VYc1SrVueobbVK1VVYqyuJUbcZ7b619c19VtS7yD6Yt1U4W7eZ6opPIkBp1loZGHZjpoEibTB2+GWqdm9GGP1k+eN4+Gw9GnW7QC2T510HoQBdurRedbxhijAIJGmRQyGGIU0So+WwjRICS3A6m5CoiLXaFQ8xT29BL0SMiu8d9l6dtx+Y825i1qBP+JeVbUenjKTUF/Spi+zdf7I1EtuzfYk8lps2t5Td2sTKyBh/J/kt35fm/OluLwQSvpQbNmkphbHWJi9JIV2zm/i9VGUYoyVk8pr0iTkR51WdfNLXUbnsbif2LeFrWnhPn2+Cry1JhX6K2P7Ofyh1q2kvpZUtkuMstcSTCPwfgOhis9MIXvfD9y+7qmhuOOSziCZY5Aa+windYR5/ZHOEEpzjzjr1z78L79MPVm3Gax/hteZffAS63p1c=</latexit>

= Aire(R)<latexit sha1_base64="Ya8HQ8izQ3EYdWbIXmvM43RVsx8=">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</latexit> ⇥ z¯

(20)

Volume =

ZZ

R

f (x, y)dV

<latexit sha1_base64="R2HAeliY84shi3X+5GnIGBJJLsE=">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</latexit>

= Aire(R)<latexit sha1_base64="Ya8HQ8izQ3EYdWbIXmvM43RVsx8=">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</latexit> ⇥ z¯

Aire(R) =

ZZ

R

dV

<latexit sha1_base64="2zUaJopd3wZV+rfA0TFLZ7H9DrA=">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</latexit>

(21)

Volume =

ZZ

R

f (x, y)dV

<latexit sha1_base64="R2HAeliY84shi3X+5GnIGBJJLsE=">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</latexit>

= Aire(R)<latexit sha1_base64="Ya8HQ8izQ3EYdWbIXmvM43RVsx8=">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</latexit> ⇥ z¯

Aire(R) =

ZZ

R

dV

<latexit sha1_base64="2zUaJopd3wZV+rfA0TFLZ7H9DrA=">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</latexit>

¯

z =

ZZ

R

f (x, y)dV ZZ

R

dV

<latexit sha1_base64="eZgu2HWcuKBoiuREBGiirEvA49U=">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</latexit>

(22)

Exemple

Trouver la valeur moyenne de la fonction f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">AAADDXicjVLLbtRAEKyYVwiPLCBx4WKxQgpCWtkLElyQIrhwDBKbRMpL9mQ2jOK1rfE4imX2G/gHrnDlhrjyDfwB3PgEapoJAiIEY3lcU93V7u7pvC5M45Lk80J05uy58xcWLy5dunzl6vLg2vX1pmqt0hNVFZXdzLNGF6bUE2dcoTdrq7NZXuiN/PCpt28caduYqnzhulrvzLKD0kyNyhypvcHN6crx3cfbU5upvpv3x7vje93ueL43GCajRFZ8GqQBDBHWWjX4hm3so4JCixk0SjjiAhkaPltIkaAmt4OenCUyYteYY4nall6aHhnZQ+4HPG0FtuTZx2xErfiXgq+lMsYdair6WWL/t1jsrUT27N9i9xLT59bxm4dYM7IOL8n+S3fi+b86X4vDFI+kBsOaamF8dSpEaaUrPvP4l6ocI9TkPN6n3RIrUZ70ORZNI7X73mZi/yKenvVnFXxbfA1ZahxJ1O5n9r3coaG9ll52RI673BJHIv1zAE6D9fEovT9Knz8Yrj4Jw7GIW7iNFU7AQ6ziGdYwYTav8AZv8S56Hb2PPkQff7hGC0FzA7+t6NN3ZImj0w==</latexit>

(23)

Exemple

Trouver la valeur moyenne de la fonction sur la région délimitée par

f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">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</latexit>

(24)

Exemple

Trouver la valeur moyenne de la fonction sur la région délimitée par

y = x

<latexit sha1_base64="/H87fIPb1z5G33CeRn6mhgmHZ8E=">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</latexit>

f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">AAADDXicjVLLbtRAEKyYVwiPLCBx4WKxQgpCWtkLElyQIrhwDBKbRMpL9mQ2jOK1rfE4imX2G/gHrnDlhrjyDfwB3PgEapoJAiIEY3lcU93V7u7pvC5M45Lk80J05uy58xcWLy5dunzl6vLg2vX1pmqt0hNVFZXdzLNGF6bUE2dcoTdrq7NZXuiN/PCpt28caduYqnzhulrvzLKD0kyNyhypvcHN6crx3cfbU5upvpv3x7vje93ueL43GCajRFZ8GqQBDBHWWjX4hm3so4JCixk0SjjiAhkaPltIkaAmt4OenCUyYteYY4nall6aHhnZQ+4HPG0FtuTZx2xErfiXgq+lMsYdair6WWL/t1jsrUT27N9i9xLT59bxm4dYM7IOL8n+S3fi+b86X4vDFI+kBsOaamF8dSpEaaUrPvP4l6ocI9TkPN6n3RIrUZ70ORZNI7X73mZi/yKenvVnFXxbfA1ZahxJ1O5n9r3coaG9ll52RI673BJHIv1zAE6D9fEovT9Knz8Yrj4Jw7GIW7iNFU7AQ6ziGdYwYTav8AZv8S56Hb2PPkQff7hGC0FzA7+t6NN3ZImj0w==</latexit>

(25)

Exemple

Trouver la valeur moyenne de la fonction sur la région délimitée par

y = x

<latexit sha1_base64="/H87fIPb1z5G33CeRn6mhgmHZ8E=">AAAC+HicjVI9T9xAEH1nSCCXDy6kTGNxQkp1spNIoUFCoaEkgrtDIgjZZrms8NmWvUaYU35CWmjTRbT8G/4BdPyEvJ0sCIIispbXb9/MG8/MTlykujJBcNHypqafPJ2ZfdZ+/uLlq7nO6/lBlddlovpJnublVhxVKtWZ6httUrVVlCoax6kaxger1j48VGWl82zTNIXaGUejTO/rJDKkNprlo91ON+gFsvyHIHSgC7fW8841vmIPORLUGEMhgyFOEaHis40QAQpyO5iQK4m02BW+o01tTS9Fj4jsAfcRT9uOzXi2MStRJ/xLyrek0sciNTn9SmL7N1/stUS27L9iTySmza3hN3axxmQNvpF9THfj+b86W4vBPpakBs2aCmFsdYmLUktXbOb+naoMIxTkLN6jvSRORHnTZ180ldRuexuJ/VI8LWvPifOtceWyVDiUqM1t9hO5Q017Ib1siAx3uSWORPj3ADwEg/e98EMv/PKxu/LZDccs3mIB7zgBn7CCNayjz2xG+IETnHrH3k/vl3f2x9VrOc0b3Fve+W8HKJwJ</latexit>

y<latexit sha1_base64="27adrWu11NFe5eaeAAwpB5qB4lY=">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</latexit> = x f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">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</latexit>

(26)

Exemple

Trouver la valeur moyenne de la fonction sur la région délimitée par

y = x

<latexit sha1_base64="/H87fIPb1z5G33CeRn6mhgmHZ8E=">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</latexit>

y<latexit sha1_base64="27adrWu11NFe5eaeAAwpB5qB4lY=">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</latexit> = x

x<latexit sha1_base64="OwlQSKK3DIu+lMpQoItZRed0NOA=">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</latexit> 2 + y2 = 4 f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">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</latexit>

(27)

Exemple

Trouver la valeur moyenne de la fonction sur la région délimitée par

y = x

<latexit sha1_base64="/H87fIPb1z5G33CeRn6mhgmHZ8E=">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</latexit>

y<latexit sha1_base64="27adrWu11NFe5eaeAAwpB5qB4lY=">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</latexit> = x x<latexit sha1_base64="DDkrrhmmrKf8WlQQwctm7PNckDc=">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</latexit> 2 + y2 = 9 x<latexit sha1_base64="OwlQSKK3DIu+lMpQoItZRed0NOA=">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</latexit> 2 + y2 = 4 f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">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</latexit>

(28)

Exemple

Trouver la valeur moyenne de la fonction sur la région délimitée par

y = x

<latexit sha1_base64="/H87fIPb1z5G33CeRn6mhgmHZ8E=">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</latexit>

y<latexit sha1_base64="27adrWu11NFe5eaeAAwpB5qB4lY=">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</latexit> = x x<latexit sha1_base64="DDkrrhmmrKf8WlQQwctm7PNckDc=">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</latexit> 2 + y2 = 9 x<latexit sha1_base64="OwlQSKK3DIu+lMpQoItZRed0NOA=">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</latexit> 2 + y2 = 4 0<latexit sha1_base64="ct1Sl6aIt1HzTve4ubNsk+ILglE=">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</latexit>  y f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">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</latexit>

(29)

Exemple

Trouver la valeur moyenne de la fonction sur la région délimitée par

y = x

<latexit sha1_base64="/H87fIPb1z5G33CeRn6mhgmHZ8E=">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</latexit>

y<latexit sha1_base64="27adrWu11NFe5eaeAAwpB5qB4lY=">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</latexit> = x x<latexit sha1_base64="DDkrrhmmrKf8WlQQwctm7PNckDc=">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</latexit> 2 + y2 = 9 x<latexit sha1_base64="OwlQSKK3DIu+lMpQoItZRed0NOA=">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</latexit> 2 + y2 = 4 0<latexit sha1_base64="ct1Sl6aIt1HzTve4ubNsk+ILglE=">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</latexit>  y f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">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</latexit>

(30)

Exemple

Trouver la valeur moyenne de la fonction sur la région délimitée par

y = x

<latexit sha1_base64="/H87fIPb1z5G33CeRn6mhgmHZ8E=">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</latexit>

y<latexit sha1_base64="27adrWu11NFe5eaeAAwpB5qB4lY=">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</latexit> = x x<latexit sha1_base64="DDkrrhmmrKf8WlQQwctm7PNckDc=">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</latexit> 2 + y2 = 9 x<latexit sha1_base64="OwlQSKK3DIu+lMpQoItZRed0NOA=">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</latexit> 2 + y2 = 4 0<latexit sha1_base64="ct1Sl6aIt1HzTve4ubNsk+ILglE=">AAAC/HicjVJNS8NAEH3Gr1q/qh69BIvgqSQq6FH04rGC/QAVSdKtLqZJTDZCKPofvOrVm3j1v/gP9OZPcHZcRS2iG7J5+2beZGZ2/CSUmXKcpyFreGR0bLw0UZ6cmp6ZrczNN7M4TwPRCOIwTtu+l4lQRqKhpApFO0mF1/ND0fLPdrS9dSHSTMbRvioScdTzTiLZlYGniGo5h6E4t4vjStWpObzsQeAaUIVZ9bjyikN0ECNAjh4EIijCITxk9BzAhYOEuCP0iUsJSbYLXKJM2py8BHl4xJ7RfkKnA8NGdNYxM1YH9JeQ3pSUNpZJE5NfSlj/zWZ7zpE1+1vsPsfUuRX09U2sHrEKp8T+pfvw/K9O16LQxSbXIKmmhBldXWCi5NwVnbn9pSpFERLiNO6QPSUcsPKjzzZrMq5d99Zj+zN7alafA+Ob48VkKXDBUYvP7Pt8h5LsCfeyIKRo51uikXB/DsAgaK7W3LWau7de3do2w1HCIpawQhOwgS3soo4G9+UaN7i1rqw76956eHe1hoxmAd+W9fgGyVmdag==</latexit>  y

=

Z 3⇡4

4

Z 3

2

r drd✓

<latexit sha1_base64="0eSMbne05GzxnZpiho9U3Z971Ks=">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</latexit>

A<latexit sha1_base64="wYtHjzDL4+JuLT6i7eepdxfn3OM=">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</latexit>

f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">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</latexit>

(31)

Exemple

Trouver la valeur moyenne de la fonction sur la région délimitée par

y = x

<latexit sha1_base64="/H87fIPb1z5G33CeRn6mhgmHZ8E=">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</latexit>

y<latexit sha1_base64="27adrWu11NFe5eaeAAwpB5qB4lY=">AAAC+XicjVLLSsRAECzj+73q0UtwEby4JCroRVj04lHBVWEVSWZHHcwmIZkshsVf8KpXb+LVr/EP9OYn2NOO4gPRCZnUVHd1uns6TCOVa8977HF6+/oHBoeGR0bHxicmK1PTe3lSZEI2RBIl2UEY5DJSsWxopSN5kGYyaIeR3A/PN419vyOzXCXxri5TedQOTmN1okSgDVWuL14cV6pezePl/gS+BVXYtZ1UXnCIFhIIFGhDIoYmHCFATk8TPjykxB2hS1xGSLFd4hIjpC3IS5JHQOw57ad0alo2prOJmbNa0F8iejNSupgnTUJ+GWHzN5ftBUc27G+xuxzT5FbSN7Sx2sRqnBH7l+7d8786U4vGCda4BkU1pcyY6oSNUnBXTObup6o0RUiJM7hF9oywYOV7n13W5Fy76W3A9if2NKw5C+tb4NlmKdHhqOVH9l2+Q0X2lHtZEtK08y3RSPjfB+An2Fuq+cs1f2elWt+wwzGEWcxhgSZgFXVsYRsNyuYMV7jGjdN1bp075/7N1emxmhl8Wc7DK6GrnEA=</latexit> = x x<latexit sha1_base64="DDkrrhmmrKf8WlQQwctm7PNckDc=">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</latexit> 2 + y2 = 9 x<latexit sha1_base64="OwlQSKK3DIu+lMpQoItZRed0NOA=">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</latexit> 2 + y2 = 4 0<latexit sha1_base64="ct1Sl6aIt1HzTve4ubNsk+ILglE=">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</latexit>  y

=

Z 3⇡4

4

5d✓

<latexit sha1_base64="vgdysTKu63Y/25jZKFD73HRjVsM=">AAADJnicjVJNTxRBEH2MosjnikcvEzcmnJYZFiIXEgIXj5i4QMLApqe3FzrMzkx6ekg2k/0Z/gf+A1e5ejPGm0e9+ROsLnuNSoj2ZHpev6pXU1VdaZnpykbR55ngwcPZR4/nnswvLC4tr7Serh5WRW2k6skiK8xxKiqV6Vz1rLaZOi6NEqM0U0fp5b6zH10pU+kif2vHpTodifNcD7UUlqh+a30n0bntN8nQCNkkpZ40m5PJmT93p8RWOEjshbKi32pHnYhXeBfEHrTh10HR+o4EAxSQqDGCQg5LOINARc8JYkQoiTtFQ5whpNmuMME8aWvyUuQhiL2k/ZxOJ57N6exiVqyW9JeMXkPKEC9JU5CfIez+FrK95siOvS92wzFdbmP6pj7WiFiLC2L/pZt6/q/O1WIxxDbXoKmmkhlXnfRRau6Kyzz8rSpLEUriHB6Q3RCWrJz2OWRNxbW73gq2f2VPx7qz9L41vvksFa446vhX9g3foSZ7yb0cE7K08y3RSMR/D8BdcLjRibud+M1me3fPD8ccnuMF1mgCXmEXr3GAHmXzDjd4j9vgOvgQfAw+/XQNZrzmGf5YwZcfkyivVQ==</latexit>

=

Z 3⇡4

4

Z 3

2

r drd✓

<latexit sha1_base64="0eSMbne05GzxnZpiho9U3Z971Ks=">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</latexit>

A<latexit sha1_base64="wYtHjzDL4+JuLT6i7eepdxfn3OM=">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</latexit>

f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">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</latexit>

(32)

Exemple

Trouver la valeur moyenne de la fonction sur la région délimitée par

y = x

<latexit sha1_base64="/H87fIPb1z5G33CeRn6mhgmHZ8E=">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</latexit>

y<latexit sha1_base64="27adrWu11NFe5eaeAAwpB5qB4lY=">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</latexit> = x x<latexit sha1_base64="DDkrrhmmrKf8WlQQwctm7PNckDc=">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</latexit> 2 + y2 = 9 x<latexit sha1_base64="OwlQSKK3DIu+lMpQoItZRed0NOA=">AAAC/nicjVJNS91AFD2mtvWjra916Sb4EITCI3kKdiNIu3Gp4FOLXyRxtIP5IpmI4VHof+i23bqTbvtX/Ae68yf0zO1YWqXohEzOnHvPzb13blymujZBcDniPRl9+uz52PjE5IuXr6Y6r99s1kVTJWqQFGlRbcdRrVKdq4HRJlXbZaWiLE7VVnzywdq3TlVV6yLfMG2p9rLoONdHOokMqY9n+/237X5/efGg0w16gSz/Pggd6MKttaJzg10cokCCBhkUchjiFBFqPjsIEaAkt4chuYpIi13hMyaobeil6BGRPeF+zNOOY3Oebcxa1An/kvKtqPQxR01Bv4rY/s0XeyORLfu/2EOJaXNr+Y1drIyswSeyD+luPR+rs7UYHOGd1KBZUymMrS5xURrpis3c/6sqwwglOYsPaa+IE1He9tkXTS21295GYr8ST8vac+J8G1y7LBVOJWr7J/uh3KGmvZRetkSGu9wSRyK8OwD3wWa/Fy70wvXF7sp7NxxjmMEs5jkBS1jBKtYwYDYZvuIbvntfvHPvwvvx29UbcZpp/LO8n78AznmdxA==</latexit> 2 + y2 = 4 0<latexit sha1_base64="ct1Sl6aIt1HzTve4ubNsk+ILglE=">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</latexit>  y

= 5

✓ 3⇡

4

⇡ 4

<latexit sha1_base64="UhYcofhmY8/CXtAlRf5vfMybfts=">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</latexit>

=

Z 3⇡4

4

5d✓

<latexit sha1_base64="vgdysTKu63Y/25jZKFD73HRjVsM=">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</latexit>

=

Z 3⇡4

4

Z 3

2

r drd✓

<latexit sha1_base64="0eSMbne05GzxnZpiho9U3Z971Ks=">AAADM3icjVJNTxRBEH2M8iECLnr0MmFj4mkzw5LgARKiF4+YuEDCwKantxc6zM5MenpINpP9M/wH/oNXvRlvxoMeTPgJVBe9BiQGezI9r1/Vq6mqrrTMdGWj6OtM8Ojx7Nz8wpPFp0vLK89aq8/3qqI2UvVkkRXmIBWVynSuelbbTB2URolRmqn99Oyds++fK1PpIv9ox6U6GomTXA+1FJaofmtrO9G57TfJ0AjZJKWeNBuTybE/d6cEO60fd0OThAMTDhJ7qqzot9pRJ+IV3gexB234tVu0rpBggAISNUZQyGEJZxCo6DlEjAglcUdoiDOENNsVJlgkbU1eijwEsWe0n9Dp0LM5nV3MitWS/pLRa0gZ4hVpCvIzhN3fQrbXHNmx/4rdcEyX25i+qY81ItbilNiHdFPP/9W5WiyGeMM1aKqpZMZVJ32UmrviMg9vVWUpQkmcwwOyG8KSldM+h6ypuHbXW8H2n+zpWHeW3rfGL5+lwjlHHf/JvuE71GQvuZdjQpZ2viUaifjvAbgP9tY7cbcTf9ho77z1w7GAl1jDa5qATezgPXbRo2wu8Amf8SW4DL4F34MfN67BjNe8wJ0V/L4GEDG0Tw==</latexit>

A<latexit sha1_base64="wYtHjzDL4+JuLT6i7eepdxfn3OM=">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</latexit>

f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">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</latexit>

(33)

Exemple

Trouver la valeur moyenne de la fonction sur la région délimitée par

y = x

<latexit sha1_base64="/H87fIPb1z5G33CeRn6mhgmHZ8E=">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</latexit>

y<latexit sha1_base64="27adrWu11NFe5eaeAAwpB5qB4lY=">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</latexit> = x x<latexit sha1_base64="DDkrrhmmrKf8WlQQwctm7PNckDc=">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</latexit> 2 + y2 = 9 x<latexit sha1_base64="OwlQSKK3DIu+lMpQoItZRed0NOA=">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</latexit> 2 + y2 = 4 0<latexit sha1_base64="ct1Sl6aIt1HzTve4ubNsk+ILglE=">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</latexit>  y

= 5

✓ 3⇡

4

⇡ 4

<latexit sha1_base64="UhYcofhmY8/CXtAlRf5vfMybfts=">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</latexit>

=

Z 3⇡4

4

5d✓

<latexit sha1_base64="vgdysTKu63Y/25jZKFD73HRjVsM=">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</latexit>

= 5⇡

2

<latexit sha1_base64="4j7uStkXmeZkWXpeWpx6sztop1c=">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</latexit>

=

Z 3⇡4

4

Z 3

2

r drd✓

<latexit sha1_base64="0eSMbne05GzxnZpiho9U3Z971Ks=">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</latexit>

A<latexit sha1_base64="wYtHjzDL4+JuLT6i7eepdxfn3OM=">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</latexit>

f (x) = y

x2 + y2

<latexit sha1_base64="Ch6Fk0tXj5p72aa9Garaslj9cWc=">AAADDXicjVLLbtRAEKyYVwiPLCBx4WKxQgpCWtkLElyQIrhwDBKbRMpL9mQ2jOK1rfE4imX2G/gHrnDlhrjyDfwB3PgEapoJAiIEY3lcU93V7u7pvC5M45Lk80J05uy58xcWLy5dunzl6vLg2vX1pmqt0hNVFZXdzLNGF6bUE2dcoTdrq7NZXuiN/PCpt28caduYqnzhulrvzLKD0kyNyhypvcHN6crx3cfbU5upvpv3x7vje93ueL43GCajRFZ8GqQBDBHWWjX4hm3so4JCixk0SjjiAhkaPltIkaAmt4OenCUyYteYY4nall6aHhnZQ+4HPG0FtuTZx2xErfiXgq+lMsYdair6WWL/t1jsrUT27N9i9xLT59bxm4dYM7IOL8n+S3fi+b86X4vDFI+kBsOaamF8dSpEaaUrPvP4l6ocI9TkPN6n3RIrUZ70ORZNI7X73mZi/yKenvVnFXxbfA1ZahxJ1O5n9r3coaG9ll52RI673BJHIv1zAE6D9fEovT9Knz8Yrj4Jw7GIW7iNFU7AQ6ziGdYwYTav8AZv8S56Hb2PPkQff7hGC0FzA7+t6NN3ZImj0w==</latexit>

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