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Texte intégral

(1)

cours 20

3.4 APPLICATIONS ET

AIRE DE SURFACE

(2)

Au dernier cours, nous avons vu

Coordonnées polaires.

Changement de variables en coordonnées polaires.

Une intégrale importante en probabilité.

(3)

Aujourd’hui, nous allons voir

Applications de l’intégrale double.

L’aire d’une surface.

(4)

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¯

¯

y = 1 b a

Z b

a

f (x)dx

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

(5)

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>

(6)

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 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=">AAADBnicjVJNT9tAEH240KZASyjHXqxGSJwiO4DgUgnRC8cgkQ+JIGRvNnQVx7bWa6TIyp3/0Gu59lb1yt/gH8CNn8DsdKlKUQVref32zbzxzOzEeaIKEwTXc96r+YXXb2pvF5eW371fqa9+6BZZqYXsiCzJdD+OCpmoVHaMMons51pGkziRvXj8xdp751IXKkuPzDSXJ5PoLFUjJSJD1Gl95fNgpCNRbQ9yNatas9N6I2gGvPynIHSgAbfaWf0OAwyRQaDEBBIpDOEEEQp6jhEiQE7cCSriNCHFdokZFklbkpckj4jYMe1ndDp2bEpnG7NgtaC/JPRqUvpYJ01Gfpqw/ZvP9pIjW/Z/sSuOaXOb0jd2sSbEGnwl9jndg+dLdbYWgxF2uQZFNeXM2OqEi1JyV2zm/l9VGYqQE2fxkOyasGDlQ5991hRcu+1txPYb9rSsPQvnW+LWZSlxzlGnf7Kv+A4V2XPu5ZSQoZ1viUYi/HcAnoJuqxluNsPDrcbevhuOGj7iEzZoAnawhwO00eEp+YbvuPQuvB/eT+/Xb1dvzmnW8Gh5V/dP76FF</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>

(7)

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⇡

2

<latexit sha1_base64="4j7uStkXmeZkWXpeWpx6sztop1c=">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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>

¯

z = 2 5⇡

Z 3⇡4

4

Z 3

2

r sin ✓

r2 r drd✓

<latexit sha1_base64="LYiZcHkLJVgQHDHpdVeFNL0nlPg=">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</latexit>

= 2 5⇡

Z 3⇡4

4

Z 3

2

sin ✓ drd✓

<latexit sha1_base64="87YPbNL5CiBLWWWU2wbx83AAJ4s=">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</latexit>

= 2 5⇡

Z 3⇡4

4

5 sin ✓ d✓

<latexit sha1_base64="FYmUavK3PrbjeVQPIMqlKEUNTmE=">AAADQXicjVLPT9RAFP4ogogCKx69NGxMPG1afkQvJkQvxhMmLpBQ3ExnZ2FCt22mU5JN0//J/8H/wYQTHuVGvHrxzdtZIhCj03T6ve+97/W9N5OWma5sFF3MBfMPFhYfLj1afvxkZXWt83R9vypqI1VfFllhDlNRqUznqm+1zdRhaZQYp5k6SM/eOf/BuTKVLvJPdlKq47E4yfVIS2GJGnQ+vElGRshms212klK3ic7toJlyzm622/azt7dmxE6YVDpP7KmyIgnD4RQNOt2oF/EK74PYgy782is610gwRAGJGmMo5LCEMwhU9BwhRoSSuGM0xBlCmv0KLZZJW1OUoghB7BntJ2QdeTYn2+WsWC3pLxm9hpQhXpCmoDhD2P0tZH/NmR37t9wN53S1Teib+lxjYi1Oif2Xbhb5vzrXi8UIr7kHTT2VzLjupM9S81Rc5eEfXVnKUBLn8JD8hrBk5WzOIWsq7t3NVrD/B0c61tnSx9a48lUqnHPWyU31DZ+hJn/Js5wQsrTzKdGViO9egPtgf7MXb/Xij9vd3bf+cizhOTbwkm7AK+ziPfbQp2q+4Bsu8T34GlwF18HPaWgw5zXPcGsFv34DYkW6mw==</latexit>

= 2

cos 3⇡

4 + cos ⇡ 4

<latexit sha1_base64="1RSmwH/FsMPURFMtOqJWmt5YkAQ=">AAADOHicjVJNT9tAEH0YaGkKJS3HXiyiSiBEZJNI7QWJj0uPVGoACSNkL5uwwrGt9RopsvJ7+A/9D1ypeuutcKv6Czo7WfqFKljL67dv5o1nZicpUlWaIPgy5U3PzD55Oves8Xx+4cVi8+Wr/TKvtJA9kae5PkziUqYqkz2jTCoPCy3jYZLKg+R819oPLqQuVZ59NKNCHg/jQab6SsSGqJPm9mbU17GoN8Z1VKhxlMq+WVmPRF5O+I5l6+547TflGD/SanBmVk+araAd8PLvg9CBFtzay5vfEeEUOQQqDCGRwRBOEaOk5wghAhTEHaMmThNSbJcYo0HairwkecTEntM+oNORYzM625glqwX9JaVXk9LHG9Lk5KcJ27/5bK84smX/F7vmmDa3EX0TF2tIrMEZsQ/p7jwfq7O1GPTxjmtQVFPBjK1OuCgVd8Vm7v9RlaEIBXEWn5JdExasvOuzz5qSa7e9jdl+w56WtWfhfCvcuiwlLjjq6Ff2Nd+hInvBvRwRMrTzLdFIhP8OwH2wv9EOO+3wQ7e1teOGYw6vsYwVmoC32MJ77KFH2VziCtf47H3yvnrfvNuJqzflNEv4a3k/fgIRaLZl</latexit>

= 2p 2

<latexit sha1_base64="fJ9d2yNQO4qWY+z0+tzGCgcplZA=">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</latexit>

(8)

Faites les exercices suivants

p.858 # 33 et 34

(9)

Pour trouver la masse d’une plaque occupant une région R<latexit sha1_base64="RDjrqD/CgSmaJTFw8Jx6WvBZVt4=">AAADIXicjVLLSsQwFD1TX+Pb0aWb4iC4GloVdDnoxuUojgoq0saowU5b0lQYBr/ArX6BX+NO3Ik/oH/hzTWCD3yktD0595yb3JvEeaIKEwSPFa+vf2BwqDo8Mjo2PjE5VZveKbJSC9kWWZLpvTgqZKJS2TbKJHIv1zLqxIncjc/XbXz3QupCZem26ebysBOdpupEicgQtbl1NFUPGgEP/zsIHajDjVZWqwzgAMfIIFCiA4kUhnCCCAU9+wgRICfuED3iNCHFcYlLjJC3JJUkRUTsOX1Pabbv2JTmNmfBbkGrJPRqcvqYJ09GOk3YruZzvOTMlv0pd49z2r116R+7XB1iDc6I/cv3rvyvz9ZicIJVrkFRTTkztjrhspTcFbtz/0NVhjLkxFl8THFNWLDzvc8+ewqu3fY24vgzKy1r58JpS7z8Wp11XLgV7K6sUjKjuVNvdfb4tBVpc+56l5ChL58n3Z7w6135DnYWG+FSI9xcrjfX3D2qYhZzWKC7soImNtBCm/d9hWvceLfenXfvPbxJvYrzzODT8J5eAaLfoyg=</latexit>

M =

ZZ

R

dm

<latexit sha1_base64="VsXz81e0EkJMlm1Nl4u4pq1JySw=">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</latexit>

Mais si la masse est répartie selon une fonction de densité

⇢(x, y) = lim

A!0

m A

<latexit sha1_base64="v3EPAw7C5U9TmNQ3dKNicllFj8Y=">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</latexit>

M =

ZZ

R

⇢(x, y)dA

<latexit sha1_base64="sqMeV5kHz2gZk4rc+MGg73uPgm4=">AAADOHicjVJdT9RAFD1bRRG/Fkl88aVxQ4KJ2bRCoi8mCC+8mKBxgYQlm7Y7wIRu20ynxM3qn+EVfgH/hDfelFd/gWcuQ6ISlWnanjn3nDtz70xa5bq2UXTWCm7dnrpzd/rezP0HDx89bs8+2ajLxmSql5V5abbSpFa5LlTPapurrcqoZJTmajM9WHXxzUNlal0Wn+y4UjujZK/QuzpLLKlB++n7t32tCzv4GPbNfrnw+eX4RTh8N2h3om4kI7wOYg868GO9nG1NoY8hSmRoMIJCAUucI0HNZxsxIlTkdjAhZ4i0xBW+YobehipFRUL2gN89zrY9W3DuctbizrhKztfQGWKenpI6Q+xWCyXeSGbH/i33RHK6vY35T32uEVmLfbL/810pb+pztVjs4o3UoFlTJYyrLvNZGumK23n4S1WWGSpyDg8ZN8SZOK/6HIqnltpdbxOJfxelY90889oGF/+szjkO/QpuV06phDHSqcs6J3LamtpKuj4msvzKefL2xH/eletg41U3XuzGH5Y6yyv+Hk3jGZ5jgXflNZaxhnX0uI8vOMIxToLT4Dz4FlxcSoOW98zhtxH8+Am2uqsc</latexit>

(10)

Exemple

Trouver la masse d’un disque de rayon 2 dont la densité est donnée par

⇢(x, y) = 1

1 + x2 + y2

<latexit sha1_base64="x2F83sEhhipu8S1pU9dlhrL2CxU=">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</latexit>

M =

Z 2⇡

0

Z 2

0

r

1 + r2 drd✓

<latexit sha1_base64="XVaE3DoZeHPLeAzf50hbO6TeZ7c=">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</latexit>

=

Z 2⇡

0

ln |1 + r2| 2

2 0

d✓

<latexit sha1_base64="WwDtvlmS9bEEXokmhqYHxBduoSI=">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</latexit>

=

Z 2⇡

0

ln 5

2 d✓

<latexit sha1_base64="2oQujGM3YGUO9Ern4gg4wFn2Pkg=">AAADRnicjVLLTtwwFD0TCkxpS4ey7CbqFKmrUTIUwaYSajddUqkDSARGjscDFpkkchykUTQ/0K9hC1/AL/QPyqa7qtte3zFSW9SHoyTH555z7XvttMx0ZaPocytYeLC4tNx+uPLo8ZPVp521Z/tVURupBrLICnOYikplOlcDq22mDkujxCTN1EF6/s7FDy6UqXSRf7TTUh1PxGmux1oKS9Sw8/JNonM7jE6aflLqWTI2QjZJlodbs6Y/GyX2TFkx7HSjXsQjvA9iD7rwY69Yay0iwQgFJGpMoJDDEs4gUNFzhBgRSuKO0RBnCGmOK8ywQt6aVIoUgthz+p7S7MizOc1dzordklbJ6DXkDLFBnoJ0hrBbLeR4zZkd+6fcDed0e5vSP/W5JsRanBH7L9+d8n99rhaLMXa4Bk01lcy46qTPUnNX3M7Dn6qylKEkzuERxQ1hyc67Pofsqbh211vB8S+sdKybS6+tcfvX6pzjwq/gduWUihnDnZrX2fBpa9KW3PUpIUtfPk+6PfHvd+U+2O/34s1e/OF1d/etv0dtPMcLvKK7so1dvMceBrSPT7jEFa6Dm+Br8C34PpcGLe9Zxy9jAT8A9qWxEQ==</latexit>

=<latexit sha1_base64="iPIOddwjldPft8cFAgE6FbujL6A=">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</latexit> ⇡ ln 5

u<latexit sha1_base64="NaaNWO2QuBbtpUIPM6Prin+pg/w=">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</latexit> = 1 + r2 du<latexit sha1_base64="hM4s5TB8OPcdwg7MS2g+46NkbQw=">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</latexit> = 2rdr

(11)

En algèbre linéaire, on a vu comment trouver le barycentre d’un ensemble de points.

# »

OB = 1 n

Xn

k=1

# » OP k

<latexit sha1_base64="PPOIhFguIwtklrDOyWiRLOhlYHg=">AAADTHicjVLLTttAFD0xBVJoIbTLbqxGlRCLyKaVYBMJ0Q27plIDSAQie5jQUfzSeBwpsvwPfA3b9gvY9w/afXdVpd65GaS2qI+xbJ8595w7c+9MXCSqNEHwqeUtPVheWW0/XFt/9Hhjs7P15LjMKy3kUORJrk/jqJSJyuTQKJPI00LLKI0TeRJPX9v4yUzqUuXZOzMv5HkaXWVqokRkiBp3dkazWf3msOn7o4mORB02ddaMyiod19N+2FxkPgsGzXg67nSDXsDDvw9CB7pwY5BvtZYxwiVyCFRIIZHBEE4QoaTnDCECFMSdoyZOE1Icl2iwRt6KVJIUEbFT+l7R7MyxGc1tzpLdglZJ6NXk9PGCPDnpNGG7ms/xijNb9k+5a85p9zanf+xypcQavCf2X7475f/6bC0GE+xzDYpqKpix1QmXpeKu2J37P1VlKENBnMWXFNeEBTvv+uyzp+TabW8jjn9mpWXtXDhthS9/rc46Zm4FuyurlMxo7tSizppPW5G24K7PCRn68nnS7Ql/vyv3wfFuL3zZC9++6h4cunvUxjM8xzbdlT0c4AgDDGkf17jBB3z0br2v3jfv+0LqtZznKX4ZSys/ABXns80=</latexit>

Mais qu’arrive-t-il si le nombre de points est infini?

(12)

1 nm

Xn

i=1

Xm

j=1

(xi, yj)

<latexit sha1_base64="yLhjMkEVDGDNR8Qmbn8q+dPgNU4=">AAADUHicjVJdT9RAFD3bFYRFYIFHXho3JJiQTQsk+mJC8MVHTFwgYbFph1kc6FemU+Km2X/hr/FVf4Fv/AP9B77BnbtDohKRadqeOfecO3PvTFKmqjJBcN3y2k9mZp/OzXcWni0uLXdXVg+rotZCDkSRFvo4iSuZqlwOjDKpPC61jLMklUfJ5RsbP7qSulJF/t6MS3maxee5GikRG6Kibn840rFowkmTZ5NhVWdRo16Hkw/5FF9YnG1+itSWP44uXkTdXtAPePj3QehAD24cFCutGQxxhgICNTJI5DCEU8So6DlBiAAlcadoiNOEFMclJuiQtyaVJEVM7CV9z2l24tic5jZnxW5Bq6T0anL62CBPQTpN2K7mc7zmzJb9V+6Gc9q9jemfuFwZsQYfif2f7075WJ+txWCEV1yDoppKZmx1wmWpuSt25/5vVRnKUBJn8RnFNWHBzrs+++ypuHbb25jjP1hpWTsXTlvj54PVWceVW8HuyiolM5o7Na2z4dNWpC2562NChr58nnR7wr/vyn1wuN0Pd/rhu93e3r67R3NYx3Ns0l15iT28xQEGtI/P+IKv+OZ99355N+3WVOq5P9bwx2h3bgFHW7Q0</latexit>

=

0

@ 1 nm

Xn i=1

Xm j=1

xi, 1 nm

Xn i=1

Xm j=1

yj

1 A

<latexit sha1_base64="g84tikZgQssRx7hereQJi1gbFug=">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</latexit>

=

0

@ 1

nm A

Xn i=1

Xm j=1

xi A, 1

nm A

Xn i=1

Xm j=1

yj A 1 A

(13)

1 nm

Xn

i=1

Xm

j=1

(xi, yj)

<latexit sha1_base64="yLhjMkEVDGDNR8Qmbn8q+dPgNU4=">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</latexit>

=

0

@ 1

nm A

Xn i=1

Xm j=1

xi A, 1

nm A

Xn i=1

Xm j=1

yj A 1 A

<latexit sha1_base64="tH00vD9kTKXkXV4L8z2GI10nWsU=">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</latexit>

=

0

@ 1 A

Xn i=1

Xm j=1

xi A, 1 A

Xn i=1

Xm j=1

yj A 1 A

<latexit sha1_base64="l0DyhcFp+P49FgjfM0z1OFzBoEU=">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</latexit>

=

0

@ 1 A

Xn i=1

Xm j=1

xi xi yj, 1 A

Xn i=1

Xm j=1

yj xi yj

1 A

<latexit sha1_base64="iKDe678n9yo8EIkJWtmpp6BSI0U=">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</latexit>

nlim!1 m!1

1 nm

Xn i=1

Xm j=1

(xi, yj) =

0 BB

@

ZZ

R

x dxdy ZZ

R

dxdy ,

ZZ

R

y dxdy ZZ

R

dxdy

1 CC A

<latexit sha1_base64="C9AsMUAlW0osdKhujSNxi7MfzxA=">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</latexit>

(14)

En jumelant cette idée et celle de la masse, on peut trouver le centre de masse.

¯

x =

ZZ

R

x⇢(x, y) dxdy ZZ

R

⇢(x, y) dxdy

<latexit sha1_base64="6JWrbPZGzMqoAPlzlfbX2E3ZzHQ=">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</latexit>

¯

y =

ZZ

R

y⇢(x, y) dxdy ZZ

R

⇢(x, y) dxdy

<latexit sha1_base64="itBIir1XUu4aWUd3Bw1McFK+ugc=">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</latexit>

(15)

Exemple

Une plaque à une forme d’un secteur circulaire de rayon 3 et d’angle d’ouverture .

3

<latexit sha1_base64="Y0ETG6N6nYwxbW01znQi2yQNRM8=">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</latexit>

Trouver son centre de masse si sa densité en chaque point est 2 fois le carré de sa distance au centre.

⇢(x, y) = 2(x2 + y2)

<latexit sha1_base64="m5JHV35edXA9VEpO1cuMNrBwmcQ=">AAADOHicjVJNT9VAFD2viiCKPDRx46bxheQRyUv7MNGNCdGNS0x8QMJX2jLAhL62mU4JzdM/41Z/gf/EnTtl6y/wzGVIVAIyTdsz555zZ+6dSatc1zaKvnWCW7en7kzP3J29d3/uwXx34eF6XTYmU6OszEuzmSa1ynWhRlbbXG1WRiXjNFcb6fEbF984UabWZfHetpXaGSeHhT7QWWJJ7XUfb5ujsn+6HLZLr8Jh/3R3+KzdHS7tdXvRIJIRXgaxBz34sVYudKawjX2UyNBgDIUCljhHgprPFmJEqMjtYELOEGmJK3zELL0NVYqKhOwxv4ecbXm24NzlrMWdcZWcr6EzxCI9JXWG2K0WSryRzI69KvdEcrq9tfynPteYrMUR2f/5LpQ39blaLA7wUmrQrKkSxlWX+SyNdMXtPPyjKssMFTmH9xk3xJk4L/ociqeW2l1vE4n/FKVj3Tzz2gZn11bnHCd+Bbcrp1TCGOnUeZ0TOW1NbSVdb4ksv3KevD3xv3flMlgfDuKVQfzueW/1tb9HM3iCp+jzrrzAKt5iDSPu4wM+4TO+BF+D78GP4OxcGnS85xH+GsGv3xFAqig=</latexit>

=

Z 3

0

r4 2

3

0d✓

<latexit sha1_base64="4hT08BmCUnvQJMK1CJ8h4mdvAuQ=">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</latexit>

=

Z 3

0

34

2 d✓

<latexit sha1_base64="dsmTE+MW6FwVle4j0iutzQpdS3c=">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</latexit>

= 27⇡

2

<latexit sha1_base64="wtY59OxAjrf3BmJZrt73sklxcS0=">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</latexit>

=

Z 3

0

Z 3

0

2r2 rdrd✓

<latexit sha1_base64="nLkAVvdFcE2emJ/8vtBc6GmAvhE=">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</latexit>

M<latexit sha1_base64="10kLUkC455/5zC+gSC0KF5DhKxM=">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</latexit>

= 2r<latexit sha1_base64="nzZaV0/XqT/lCsUqFcLfLQeNpEQ=">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</latexit> 2

(16)

Exemple

Une plaque à une forme d’un secteur circulaire de rayon 3 et d’angle d’ouverture .

3

<latexit sha1_base64="Y0ETG6N6nYwxbW01znQi2yQNRM8=">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</latexit>

Trouver son centre de masse si sa densité en chaque point est 2 fois le carré de sa distance au centre.

⇢(x, y) = 2(x2 + y2)

<latexit sha1_base64="m5JHV35edXA9VEpO1cuMNrBwmcQ=">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</latexit>

= 27⇡

2

<latexit sha1_base64="wtY59OxAjrf3BmJZrt73sklxcS0=">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</latexit>

M<latexit sha1_base64="10kLUkC455/5zC+gSC0KF5DhKxM=">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</latexit>

x = r cos ✓

<latexit sha1_base64="idf8aSH+0/mjJ/9odZhEud6xNNI=">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</latexit>

= 2 27⇡

Z 3

0

Z 3

0

2r4 cos ✓ drd✓

<latexit sha1_base64="dKcNlmT4CbapdKTWvNMh+QPu06Q=">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</latexit>

¯ x<latexit sha1_base64="lFGg+vYDe0qypsT7MrfqM8NIUL0=">AAADJnicjVLLTtwwFD0TyrOU8liyiTqqxGqUtEh0iWDTJZWYAQlQlQRDXTJJ5DiI0Yh/YAtf0K9hh6ruuoS/4PhiJB6i1FGS43PPufa9dlrlurZR9KcVjLwZHRufmJx6O/1u5v3s3HyvLhuTqW5W5qXZTpNa5bpQXattrrYro5J+mqut9GjdxbeOlal1WWzaQaX2+slhoQ90llhSvd00MeHJ99l21IlkhM9B7EEbfmyUc61R7GIfJTI06EOhgCXOkaDms4MYESpyexiSM0Ra4gqnmKK3oUpRkZA94veQsx3PFpy7nLW4M66S8zV0hvhIT0mdIXarhRJvJLNjX8o9lJxubwP+U5+rT9biB9nXfPfK//W5WiwO8EVq0KypEsZVl/ksjXTF7Tx8UJVlhoqcw/uMG+JMnPd9DsVTS+2ut4nEr0XpWDfPvLbBzT+rc45jv4LblVMqYYx06q7OoZy2praSrg+ILL9ynrw98dO78hz0PnXiz53423J7dc3fowks4gOWeFdWsIqv2ECX+/iJM5zjIvgVXAZXwe87adDyngU8GsHfW3EUpTE=</latexit>

= 4 · 32 5⇡

⇣sin ⇡

3 sin 0⌘

<latexit sha1_base64="Wj3FwkAsdDMRScGfpv/B4x2v3Vo=">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</latexit>

= 18p 3 5⇡

<latexit sha1_base64="PRxRNq/Hf0QGuCnLZH87i5WrvZY=">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</latexit>

= 4 33

Z 3

0

35

5 cos ✓ d✓

<latexit sha1_base64="Ewmhza+clSuBUI7fp2AKjqkAlSw=">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</latexit>

= 2r<latexit sha1_base64="nzZaV0/XqT/lCsUqFcLfLQeNpEQ=">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</latexit> 2

(17)

Exemple

Une plaque à une forme d’un secteur circulaire de rayon 3 et d’angle d’ouverture .

3

<latexit sha1_base64="Y0ETG6N6nYwxbW01znQi2yQNRM8=">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</latexit>

Trouver son centre de masse si sa densité en chaque point est 2 fois le carré de sa distance au centre.

⇢(x, y) = 2(x2 + y2)

<latexit sha1_base64="m5JHV35edXA9VEpO1cuMNrBwmcQ=">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</latexit>

= 27⇡

2

<latexit sha1_base64="wtY59OxAjrf3BmJZrt73sklxcS0=">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</latexit>

M<latexit sha1_base64="10kLUkC455/5zC+gSC0KF5DhKxM=">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</latexit>

¯

x<latexit sha1_base64="lFGg+vYDe0qypsT7MrfqM8NIUL0=">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</latexit> = 18p 3 5⇡

<latexit sha1_base64="PRxRNq/Hf0QGuCnLZH87i5WrvZY=">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</latexit>

¯ y

<latexit sha1_base64="RWrCRvzL9/HhxaIn5yuEU9crk9g=">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</latexit>

y = r sin ✓

<latexit sha1_base64="BsTvw+6rzg4SGWRRbUQ1QzZKpao=">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</latexit>

= 2 27⇡

Z 3

0

Z 3

0

2r4 sin ✓ drd✓

<latexit sha1_base64="lgfgxbf4PrtQnItth8qkpnKXsmM=">AAADanicjVJNT9tAEH2JS0vpBwFOFReraaWeIjtBohck1F44UqkBJAzRerOBFY5trddIkZW/xK/hwqE9txI/orOTjdQW9WMt22/fvDe7M7tpmenKRtFdqx08Wnn8ZPXp2rPnL16udzY2j6qiNlINZZEV5iQVlcp0roZW20ydlEaJaZqp4/Tqo4sfXytT6SL/bGelOpuKi1xPtBSWqFHnYC+ZGCGb/rzp7yalnic6t6PovFnQjmkG8yU7CPvmfCepdJ7YS2VFEo7NeAFHnW7Ui3iED0HsQRd+HBYbrRUkGKOARI0pFHJYwhkEKnpOESNCSdwZGuIMIc1xhTnWyFuTSpFCEHtF3wuanXo2p7nLWbFb0ioZvYacId6SpyCdIexWCzlec2bH/il3wznd3mb0T32uKbEWl8T+y7dU/q/P1WIxwXuuQVNNJTOuOumz1NwVt/Pwp6osZSiJc3hMcUNYsnPZ55A9Fdfueis4/o2VjnVz6bU1vv+1Oue49iu4XTmlYsZwpxZ1NnzamrQld31GyNKXz5NuT/z7XXkIjvq9eNCLP+109z/4e7SKbbzGO7oru9jHAQ4xpH3c4BZf8LV9H2wGr4LthbTd8p4t/DKCNz8A2eu8/w==</latexit>

= 36 5⇡

Z 3

0

sin ✓ d✓

<latexit sha1_base64="RvB7h8lIpcjpoh/57dVavn9iHmg=">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</latexit>

= 36 5⇡

⇣ cos ⇡

3 + cos 0⌘

<latexit sha1_base64="oYECL4aXQbvCMCDr3jJ18boIdSM=">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</latexit>

= 18 5⇡

<latexit sha1_base64="OUUVjYXuLhmv4f+iXoRZEu0D+hU=">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</latexit>

= 2r<latexit sha1_base64="nzZaV0/XqT/lCsUqFcLfLQeNpEQ=">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</latexit> 2

(18)

Exemple

Une plaque à une forme d’un secteur circulaire de rayon 3 et d’angle d’ouverture .

3

<latexit sha1_base64="Y0ETG6N6nYwxbW01znQi2yQNRM8=">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</latexit>

Trouver son centre de masse si sa densité en chaque point est 2 fois le carré de sa distance au centre.

⇢(x, y) = 2(x2 + y2)

<latexit sha1_base64="m5JHV35edXA9VEpO1cuMNrBwmcQ=">AAADOHicjVJNT9VAFD2viiCKPDRx46bxheQRyUv7MNGNCdGNS0x8QMJX2jLAhL62mU4JzdM/41Z/gf/EnTtl6y/wzGVIVAIyTdsz555zZ+6dSatc1zaKvnWCW7en7kzP3J29d3/uwXx34eF6XTYmU6OszEuzmSa1ynWhRlbbXG1WRiXjNFcb6fEbF984UabWZfHetpXaGSeHhT7QWWJJ7XUfb5ujsn+6HLZLr8Jh/3R3+KzdHS7tdXvRIJIRXgaxBz34sVYudKawjX2UyNBgDIUCljhHgprPFmJEqMjtYELOEGmJK3zELL0NVYqKhOwxv4ecbXm24NzlrMWdcZWcr6EzxCI9JXWG2K0WSryRzI69KvdEcrq9tfynPteYrMUR2f/5LpQ39blaLA7wUmrQrKkSxlWX+SyNdMXtPPyjKssMFTmH9xk3xJk4L/ociqeW2l1vE4n/FKVj3Tzz2gZn11bnHCd+Bbcrp1TCGOnUeZ0TOW1NbSVdb4ksv3KevD3xv3flMlgfDuKVQfzueW/1tb9HM3iCp+jzrrzAKt5iDSPu4wM+4TO+BF+D78GP4OxcGnS85xH+GsGv3xFAqig=</latexit>

= 27⇡

2

<latexit sha1_base64="wtY59OxAjrf3BmJZrt73sklxcS0=">AAADMnicjVJda9RAFD0brdb6ta2PvgQXwaclWQv1RSj60scKblvolpJMZ+vQbBImk4Ul7D/xVX+Bf0bfpC8V/BGeuZ2CWvyYkOTMuefcmXtn8rowjUuSz73oxs2VW7dX76zdvXf/wcP++sZeU7VW6bGqisoe5FmjC1PqsTOu0Ae11dksL/R+fvbax/fn2jamKt+6Ra2PZtlpaaZGZY7Ucb//cjK1mepGW5PaLLvR8rg/SIaJjPg6SAMYIIzdar23gglOUEGhxQwaJRxxgQwNn0OkSFCTO0JHzhIZiWsssUZvS5WmIiN7xu8pZ4eBLTn3ORtxK65S8LV0xnhKT0WdJfarxRJvJbNn/5S7k5x+bwv+85BrRtbhHdl/+a6U/+vztThM8UJqMKypFsZXp0KWVrridx7/VJVjhpqcxyeMW2Ilzqs+x+JppHbf20ziF6L0rJ+roG3x7a/Vecc8rOB35ZVaGCuduqyzk9M21NbS9QWR41fOk7cn/f2uXAd7o2H6fJi+2Rxsvwr3aBWP8QTPeFe2sI0d7GLMfczxHh/wMfoUfYm+RueX0qgXPI/wy4i+/wAYvKmS</latexit>

M<latexit sha1_base64="10kLUkC455/5zC+gSC0KF5DhKxM=">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</latexit>

¯

x<latexit sha1_base64="lFGg+vYDe0qypsT7MrfqM8NIUL0=">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</latexit> = 18p 3 5⇡

<latexit sha1_base64="PRxRNq/Hf0QGuCnLZH87i5WrvZY=">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</latexit>

¯ y

<latexit sha1_base64="RWrCRvzL9/HhxaIn5yuEU9crk9g=">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</latexit>

y<latexit sha1_base64="BsTvw+6rzg4SGWRRbUQ1QzZKpao=">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</latexit> = r sin ✓

= 18 5⇡

<latexit sha1_base64="OUUVjYXuLhmv4f+iXoRZEu0D+hU=">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</latexit>

= 2r<latexit sha1_base64="nzZaV0/XqT/lCsUqFcLfLQeNpEQ=">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</latexit> 2

(19)

Faites les exercices suivants

p.866 # 3 à 10 p.867 # 3 à 10

(20)

Aire d’une surface

Références

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