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3.3 COORDONNÉES POLAIRES

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Texte intégral

(1)

cours 19

3.3 COORDONNÉES

POLAIRES

(2)

Au dernier cours, nous avons vu

Intégrale double sur des régions non rectangulaires.

Région de Type I et II

(3)

Aujourd’hui, nous allons voir

Coordonnées polaires.

Changement de variables en coordonnées polaires.

Une intégrale importante en probabilité.

(4)

x

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y

<latexit sha1_base64="ADnGsQtcomAbrlTQt0snNnnV3IQ=">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</latexit>

r

<latexit sha1_base64="7UIp2xsOnAsSXwKDOoBrzyEgGF8=">AAAC9nicjVJNS8NAEH2N399Vj16CRfBUUhX0WPTi0YKtQi2SpGtdmiZhsymW4i/wqldv4tW/4z/Qmz/B2XEVPxDdkM3bN/MmM7MTpJHMtOc9FpyR0bHxicmp6ZnZufmF4uJSI0tyFYp6mESJOg78TEQyFnUtdSSOUyX8XhCJo6C7Z+xHfaEymcSHepCKVs/vxPJMhr4mqqZOiyWv7PFyf4KKBSXYdZAUX3CCNhKEyNGDQAxNOIKPjJ4mKvCQEtfCkDhFSLJd4BLTpM3JS5CHT2yX9g6dmpaN6WxiZqwO6S8RvYqULtZIk5CfImz+5rI958iG/S32kGOa3Ab0DWysHrEa58T+pXv3/K/O1KJxhh2uQVJNKTOmutBGybkrJnP3U1WaIqTEGdwmuyIcsvK9zy5rMq7d9NZn+xN7GtacQ+ub49lmKdDnqIOP7Id8h5LsKfdyQEjTzrdEI1H5PgA/QWOjXNkse7WtUnXXDsckVrCKdZqAbVSxjwPUOZsrXOPGuXBunTvn/s3VKVjNMr4s5+EV1lybOA==</latexit>

<latexit sha1_base64="oMWpbHW/73RUgpyF2ZqjqN+mpCA=">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</latexit>

x = r cos ✓

<latexit sha1_base64="iEUrtxMHm2EsciY7SRWG265yVO4=">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</latexit>

y = r sin ✓

<latexit sha1_base64="FLSFl8OgX1eHh8XrB3DRpt0td5c=">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</latexit>

r

<latexit sha1_base64="fWAe7a2GRrHwICXqOBj4AZb7j+I=">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</latexit> 2

= x

2

+ y

2

r = p

x

2

+ y

2

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tan ✓ = y x

<latexit sha1_base64="lFRGYnnat8P+gY7+c9dY+eUs7RE=">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</latexit>

✓ = arctan ⇣ y x

<latexit sha1_base64="qruMrD7WrwjhEzSwjhdkaxxhPKs=">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</latexit>

✓ = arctan ⇣ y x

⌘ + ⇡

<latexit sha1_base64="ipGQ3QMlADbujFmGkT5G7EX4518=">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</latexit>

x

<latexit sha1_base64="CqmjzSUxxjGGlkiWVDqIet9p89Q=">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</latexit>

0 si

x <

<latexit sha1_base64="cQH+0IONzV2JTlIzbNz8pPqU/gQ=">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</latexit>

0

si

(5)

plan cartésien plan polaire 1 < x < 1

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1 < y < 1

<latexit sha1_base64="aVjvtCGR9i1Gz0yo8/2833fGn7g=">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</latexit>

0  ✓ < 2⇡

<latexit sha1_base64="VrO14SMbwEnnbiRlXR9KgoyINFU=">AAADCHicjVKxTtxAEH04IRBCwgFlGotTJKqTTZBCQXEKDSWROEDC6GT7Fljhsx17fdHplB/IP9Ambboobf6CP4COT+DtsKAkCCVref32zbzxzOwkZaZrEwQXU96Tp9PPZmafz72Yf/lqobW4tFcXTZWqXlpkRXWQxLXKdK56RptMHZSViodJpvaTsy1r3x+pqtZFvmvGpToaxie5PtZpbEj1W4tBlKmPkTlVJt7016JS91vtoBPI8h+C0IE23NopWteIMECBFA2GUMhhiDPEqPkcIkSAktwRJuQqIi12hc+Yo7ahl6JHTPaM+wlPh47NebYxa1Gn/EvGt6LSxxtqCvpVxPZvvtgbiWzZx2JPJKbNbcxv4mINyRqckv2X7s7zf3W2FoNjbEgNmjWVwtjqUhelka7YzP3fqjKMUJKzeEB7RZyK8q7Pvmhqqd32Nhb7pXha1p5T59vgymWpMJKo4/vsJ3KHmvZSejkmMtzlljgS4d8D8BDsrXXCt53gw3q7+94NxyxeYwWrnIB36GIbO+gxm084x1d88754370f3s9bV2/KaZbxx/J+3QBPsqGa</latexit>

0

<latexit sha1_base64="rosckR68qzTRWtScWKQDgrg1xhk=">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</latexit>

 r < 1

(6)

dA

<latexit sha1_base64="3fDvLflcrYK3QXuBg6CIAk07ysc=">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</latexit>

= dxdy

dA

<latexit sha1_base64="vxp3lGylSVCP15AhtIEkb3PBYC8=">AAADAnicjVJNTxRBEH2MiIAKqxy5TNiYeNrMigleTBAvHDFhgQQ2Zj4a6DA7M/b0bLLZeOM/cMUrN8KVP+I/0Bs/gddFQ1BipCfT8/pVvZqq6kqqXNc2in5OBE8mn049m56Zff7i5dx869XrrbpsTKp6aZmXZieJa5XrQvWstrnaqYyKB0mutpOjz86+PVSm1mWxaUeV6g/ig0Lv6zS2pPrZp48mM9mePVQ2/tpqR51IVvgQdD1ow6+NsnWFPWQokaLBAAoFLHGOGDWfXXQRoSLXx5icIdJiV/iOWWobeil6xGSPuB/wtOvZgmcXsxZ1yr/kfA2VId5QU9LPELu/hWJvJLJj/xV7LDFdbiN+Ex9rQNbikOz/dLeej9W5Wiz28UFq0KypEsZVl/oojXTFZR7eq8oyQkXO4Yx2Q5yK8rbPoWhqqd31Nhb7L/F0rDun3rfBb5+lwlCiju6yH8sdator6eWIyHKXW+JIdP8egIdg612nu9yJvrxvr6754ZjGIpbwlhOwglWsYwM9ZvMNJzjFj+A4OAvOg4sb12DCaxbwxwourwFwjaA+</latexit>

= rdrd✓

dx

<latexit sha1_base64="04sGZVEbtthbG6zjtHPZS/cJILQ=">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</latexit>

dy

<latexit sha1_base64="CrCoOs5W3zo05ogWP67EdjT+JoU=">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</latexit>

d✓

<latexit sha1_base64="G//6V94EvkA5k+7pJoQ2rF8SnXA=">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</latexit>

r

<latexit sha1_base64="7UIp2xsOnAsSXwKDOoBrzyEgGF8=">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</latexit>

dr

<latexit sha1_base64="do3wYbgwiH0WBdmVAvcFD26vZe0=">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</latexit>

rd✓

<latexit sha1_base64="cw0tlCDcUDqjSuehkiTGyQ0r6zs=">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</latexit>

(7)

X

n

i=1

X

m

j=1

f (x

i

, y

j

) A

<latexit sha1_base64="9n8rbShDaGkSxVOgGYDaysr4LOQ=">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</latexit>

=

X

n

i=1

X

m

j=1

f (r

i

cos ✓

i

, r

j

sin ✓

j

) A

<latexit sha1_base64="RNzhamD5t5DSrMsing5kFjCyzo4=">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</latexit>

=

X

n

i=1

X

m

j=1

f (r

i

cos ✓

i

, r

j

sin ✓

j

)r r ✓

<latexit sha1_base64="7dgp1+AgwpfYhjg6nTKxP60Gr6A=">AAADSXicjVJNTxRBEH3MqiB+rXj0MnFjgonZzKCJXEiIevAIxgUSBic9TS/0Ml/p7iHZbPhf/Ad/gRev4k1vhpPVRWNUYrQn0/36Vb3qquou2lJblyQf56Letes35hduLt66fefuvf79pS3bdEaqkWzKxuwUwqpS12rktCvVTmuUqIpSbRdHr7x9+1gZq5v6nZu2aq8SB7UeaykcUXn/7Vpmuyqf6bX05H19gSceV/F42eQ6k43N3KFyItdPY5NPMqvrQEyemOy1Kp2Iwxr4/iAZJjziqyANYIAwNpr+V2TYRwOJDhUUajjCJQQsfbtIkaAlbg8z4gwhzXaFEyyStiMvRR6C2COaD2i3G9ia9j6mZbWkU0r6DSljPCZNQ36GsD8tZnvHkT37t9gzjulzm9JahFgVsQ6HxP5Ld+n5vzpfi8MYq1yDpppaZnx1MkTpuCs+8/iXqhxFaInzeJ/shrBk5WWfY9ZYrt33VrD9jD096/cy+Hb4ErJUOOao05/Zz/gONdlb7uWUkKOZb4meRPrnA7gKtlaG6bNhsvl8sP4yPI4FPMQjLNMLeIF1vMEGRpTNKT7hM86iD9G36Ht0fuEazQXNA/w2er0fw5y8Cw==</latexit>

=

X

n

i=1

X

m

j=1

g(r

i

, ✓

j

) r ✓

<latexit sha1_base64="hFLInwJmSZN+OEcsOHZtE1B1OQQ=">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</latexit>

nous donne une somme de Riemann

(8)

x = r cos ✓

<latexit sha1_base64="iEUrtxMHm2EsciY7SRWG265yVO4=">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</latexit>

y = r sin ✓

<latexit sha1_base64="FLSFl8OgX1eHh8XrB3DRpt0td5c=">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</latexit>

dx = @ x

@ r dr + @ x

@✓ d✓

<latexit sha1_base64="HQ0/hkISRhIDaLiQs2e8YCPi14g=">AAADQ3icjVLLahRBFD1pXzG+Rl26aRwEQRh6VNCNEHTjwkUEJwlkQqiurkmK9HQ31dUhwzA/5T/4D7rUleJGxK3gqZuKr+Cjmq46de49t+69VXlT2tZn2Zul5NTpM2fPLZ9fuXDx0uUrvavX1tu6c9qMdF3WbjNXrSltZUbe+tJsNs6oaV6ajXz/SbBvHBjX2rp64WeN2Z6q3cpOrFae1E7vWXH4aDxxSs/HjXLeqjI9XPzAblG4O3+xj/2e8WpRHK07vX42yGSkJ8Ewgj7iWKt7nzBGgRoaHaYwqOCJSyi0/LYwRIaG3Dbm5ByRFbvBAivUdvQy9FBk9znvcrcV2Yr7ELMVteYpJX9HZYpb1NT0c8ThtFTsnUQO7J9izyVmyG3GNY+xpmQ99sj+S3fs+b+6UIvHBA+lBsuaGmFCdTpG6aQrIfP0p6o8IzTkAi5od8RalMd9TkXTSu2ht0rs78UzsGGvo2+HDzFLgwOJOvue/Vzu0NLeSC9nRJ6z3BKfxPD3B3ASrN8dDO8Nsuf3+6uP4+NYxg3cxG2+gAdYxVOsYcRsXuI13uJd8ir5mHxOvhy5JktRcx2/jOTrN58YvB0=</latexit>

dy = @ y

@ r dr + @ y

@✓ d✓

<latexit sha1_base64="fKzb5dhSVRnLiBo+Fc1RlHlbmdg=">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</latexit>

= cos

<latexit sha1_base64="kKF0Mg1WSC1JG7i0W9d1sBxwQ3k=">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</latexit>

✓dr r sin ✓d✓

= sin

<latexit sha1_base64="0uxsLyoukHtR4NOdLLoFLmL6ur8=">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</latexit>

✓dr + r cos ✓d✓

✓ dx dy

=

✓ cos ✓ r sin ✓ sin ✓ r cos ✓

◆ ✓ dr d✓

(9)

x = r cos ✓

<latexit sha1_base64="iEUrtxMHm2EsciY7SRWG265yVO4=">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</latexit>

y = r sin ✓

<latexit sha1_base64="FLSFl8OgX1eHh8XrB3DRpt0td5c=">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</latexit>

✓ dx dy

=

✓ cos ✓ r sin ✓ sin ✓ r cos ✓

◆ ✓ dr d✓

<latexit sha1_base64="LoryW1G5o/KDk2bNdr9k2LiYgK0=">AAADv3icjVJNb9QwEJ00fJTla6HHXiJWoHJglS1I7QWpwKXHInXbSvVSOY43azXrRLZTEUX7//gL/AO49ScwnrpQWFbg1SZv3sx7nnGc1aWyLk2/Rmvxrdt37q7f691/8PDR4/6Tp0e2aoyQY1GVlTnJuJWl0nLslCvlSW0kn2elPM7OP/j88YU0VlX60LW1nMx5odVUCe6QOut/YaWcui2WyULpjhvD20UnFr38M2NJ3iZM6jzQzKhi5l6+7f1VghomKsvcTDqevEheGWaVvgrR6leAOXOjcnmDVR0Z39FK1Vl/kA5TWskyGAUwgLAOqv4lMMihAgENzEGCBoe4BA4Wf6cwghRq5CbQIWcQKcpLWEAPtQ1WSazgyJ7js8DoNLAaY+9pSS1wlxL/BpUJPEdNhXUGsd8toXxDzp5d5d2Rp++txXcWvObIOpgh+y/ddeX/6vwsDqawSzMonKkmxk8ngktDp+I7T25M5dChRs7jHPMGsSDl9TknpLE0uz9bTvlvVOlZH4tQ28D30KWEC3Jtf3bf0TdUmK/pLFtEDp/0lfBKjP68AMvgaHs4ej1MP74Z7L0Pl2MdNuEZbOEN2IE92IcDGIOIhtFhNIk+xe/iItZxfVW6FgXNBvy24vYHq1Pl+w==</latexit>

Mais on a vu en algèbre linéaire que le déterminant donne le facteur de changement d’aire.

cos ✓ r sin ✓

sin ✓ r cos ✓ =

<latexit sha1_base64="cC+oduu9cgz3WpXz9TLGOT41G3o=">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</latexit>

r cos

2

✓ + r sin

2

✓ =

<latexit sha1_base64="qYB0MQGOV3uC5J0L2VOCuR/gqps=">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</latexit>

r

(10)

La matrice jacobienne.

Nous reverrons ça un peu plus tard, mais si on a un changement de variables

x

<latexit sha1_base64="JY2khH6//80l98yMZSz93g8OdfY=">AAAC/XicjVJLS8NAEP4a3/VV9eglWIQKUlIV9CIUvXhUsA+oIklc62KahGRTLEX8D1716k28+lv8B3rzJzg7ruID0Q3ZfPvNfJOZ2fHiQKbKcR5z1sDg0PDI6Fh+fGJyarowM1tPoyzxRc2Pgihpem4qAhmKmpIqEM04EW7HC0TDO9vW9kZXJKmMwn3Vi8Vhx22H8kT6riKqeb7ZLmXL3aWjQtEpO7zsn6BiQBFm7UaFFxzgGBF8ZOhAIIQiHMBFSk8LFTiIiTtEn7iEkGS7wAXypM3IS5CHS+wZ7W06tQwb0lnHTFnt018CehNS2lgkTUR+CWH9N5vtGUfW7G+x+xxT59ajr2didYhVOCX2L9275391uhaFE2xwDZJqipnR1fkmSsZd0Znbn6pSFCEmTuNjsieEfVa+99lmTcq16966bH9iT83qs298MzybLAW6HLX3kX2f71CSPeZe9ggp2vmWaCQq3wfgJ6ivlCurZWdvrVjdMsMxinksoEQTsI4qdrCLGt/8Fa5xY11at9addf/mauWMZg5flvXwCjxXnZA=</latexit>

= g(u, v ) y

<latexit sha1_base64="lSfOKP3gVoNokHeza5kzULQifzg=">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</latexit>

= h(u, v) dx = @ x

@ u du + @ x

@ v dv

<latexit sha1_base64="LK96FCXh9BAI8LkkhX0o8y2ST8o=">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</latexit>

dy = @ y

@ u du + @ y

@ v dv

<latexit sha1_base64="n+Yo2Gu0LZNMaSHm74hrtj1n37g=">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</latexit>

✓ dx dy

=

0 B B B @

@ x

@ u

@ x

@ v

@ y

@ u

@ y

@ v

1 C C C A

✓ du dv

<latexit sha1_base64="ZniGZc5J/Z2twmsA3vSEuBtGibI=">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</latexit>

(11)

Exemple

x

<latexit sha1_base64="4vSRZXcHfkzYJPzoWi6LlQ1Ev6Q=">AAADBnicjVJNT9wwEH0E2gL9YAtHLhErpEqVVtkFiV6QEL1wpIgFJMqiJBhqkU2ixEGkK+79D1zhyq3qtX+DfwA3fgLPg0G0CLWO7Dy/mTf2jCfKE12aILgc8oZHXrx8NTo2/vrN23cTjfeTG2VWFbHqxlmSFVtRWKpEp6prtEnUVl6osB8lajM6/Gztm0eqKHWWrps6Vzv98CDV+zoODandxsRxr/Ox5vze6yyu9Tq7jWbQCmT4T0HbgSbcWM0aN/iKPWSIUaEPhRSGOEGIkt822giQk9vBgFxBpMWucIJxait6KXqEZA+5HnC37diUexuzFHXMUxLOgkofs9Rk9CuI7Wm+2CuJbNnnYg8kpr1bzX/kYvXJGnwj+y/dvef/6mwuBvv4JDlo5pQLY7OLXZRKqmJv7j/KyjBCTs7iPdoL4liU93X2RVNK7ra2odivxNOydh873wrX7pYKRxK1frj9QN5Q055LLWsiw1VeiS3R/rsBnoKNTqs91wq+zDeXll1zjGIaM/jADljAElawiq50ySnOcO798C68n96vO1dvyGmm8Mfwft8COiKgEw==</latexit> 2

+ y

2

+ z

2

= R

2

z

<latexit sha1_base64="qyugzFU/WjYmS5/FXG4PKQt5TmM=">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</latexit> 2

= R

2

x

2

y

2

z = p

R

2

x

2

y

2

<latexit sha1_base64="jspS4kINDsLyA5voIJenY1/8570=">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</latexit>

z = p

R

2

x

2

y

2

<latexit sha1_base64="pjKvi9oKghEccsgr8gUAdzalGL8=">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</latexit>

Trouver le volume de la sphère

(12)

Exemple

2

Z

R

R

Z

pR2 x2

pR2 x2

p R

2

x

2

y

2

dydx

<latexit sha1_base64="8YZgC1qAlp256jtNJCxmgkVHIoc=">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</latexit>

Substitution trig!

z = p

R

2

x

2

y

2

<latexit sha1_base64="jspS4kINDsLyA5voIJenY1/8570=">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</latexit>

R

<latexit sha1_base64="30FFIJ+2gmqZgHyn3pzBmAsFzAE=">AAAC93icjVJNT9tAEH0Y2kJKSwpHLhZRJU6R3VYqx6hcegREIBJEle1s6CqObdnrSFbUf9ArvfZW9dqf038AN34Cb4el4kNVu5bXb9/MG8/MTlykujJB8HvBW1x68vTZ8krr+eqLl2vtV+tHVV6XieoneZqXgziqVKoz1TfapGpQlCqaxqk6jie71n48U2Wl8+zQNIUaTqOzTI91EhlSBwetT+1O0A1k+Y9B6EAHbu3l7SucYoQcCWpMoZDBEKeIUPE5QYgABbkh5uRKIi12hS9oUVvTS9EjIjvhfsbTiWMznm3MStQJ/5LyLan08ZqanH4lsf2bL/ZaIlv2b7HnEtPm1vAbu1hTsgafyf5Ld+v5vzpbi8EYO1KDZk2FMLa6xEWppSs2c/9OVYYRCnIWj2gviRNR3vbZF00ltdveRmK/EE/L2nPifGtcuiwVZhK1+ZP9XO5Q015ILxsiw11uiSMRPhyAx+DoTTd82w3233V6H9xwLGMTW9jmBLxHDx+xhz6zGeMrzvHNa7zv3g/v542rt+A0G7i3vF/XxBSbLA==</latexit>

y = p

R

2

x

2

<latexit sha1_base64="KG15/B/jz14YgjbOcCpo/p0acZA=">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</latexit>

y = p

R

2

x

2

<latexit sha1_base64="Gf6+5RoJ322S6ZPybQ37uRrxSOc=">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</latexit>

(13)

Exemple

2

Z

R

R

Z

pR2 x2 pR2 x2

p R

2

x

2

y

2

dydx

<latexit sha1_base64="8YZgC1qAlp256jtNJCxmgkVHIoc=">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</latexit>

z = p

R

2

x

2

y

2

<latexit sha1_base64="jspS4kINDsLyA5voIJenY1/8570=">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</latexit>

R

<latexit sha1_base64="30FFIJ+2gmqZgHyn3pzBmAsFzAE=">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</latexit>

x

<latexit sha1_base64="iEUrtxMHm2EsciY7SRWG265yVO4=">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</latexit>

= r cos ✓ y

<latexit sha1_base64="FLSFl8OgX1eHh8XrB3DRpt0td5c=">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</latexit>

= r sin ✓

0

<latexit sha1_base64="2s4HrTpuyWjipcfPbYaxj60NRRQ=">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</latexit>

 r  R

0  ✓  2⇡

<latexit sha1_base64="aZ9A7M9l63ZiA+VTqAB5G2avmqo=">AAADDHicjVJNT9tAEH0x/aApLSkVp16sRpU4RQ5UokdULhxTqYFIJEK2swkrHNu110hR1L/Q/8AVrtwQ1/6H/oP21p/A2+mmKkQVrOX12zfzxjOzE+WJLk0Q/Kh5S48eP3m6/Kz+fOXFy9XGq7X9MquKWHXjLMmKXhSWKtGp6hptEtXLCxVOokQdRCe71n5wqopSZ+lnM83VYBKOUz3ScWhIHTXWg36ivvh9c6xMKHCzn+ujRjNoBbL8RdB2oAm3OlnjN/oYIkOMChMopDDECUKUfA7RRoCc3AAzcgWRFrvCV9Spreil6BGSPeE+5unQsSnPNmYp6ph/SfgWVPp4R01Gv4LY/s0XeyWRLfu/2DOJaXOb8hu5WBOyBsdk79PNPR+qs7UYjPBBatCsKRfGVhe7KJV0xWbu/1OVYYScnMVD2gviWJTzPvuiKaV229tQ7D/F07L2HDvfCr9clgqnEnX6N/uZ3KGmPZdeTokMd7kljkT77gAsgv3NVnurFXx639z56IZjGW/wFhucgG3sYA8ddJnNDGc4x4X3zbv0rrzrP65ezWle49byvt8A4oujRA==</latexit>

= 2

Z

2⇡

0

Z

r

0

p R

2

r

2

rdrd✓

<latexit sha1_base64="i/bCt9VICn3UR8y9hiRtxhi0Bzs=">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</latexit>

u

<latexit sha1_base64="DCQj9ECCerjbriI9IX1Ynge7bGM=">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</latexit>

= R

2

r

2

=

Z

2⇡

0

2(R

2

r

2

)

32

3

r

0

d✓

<latexit sha1_base64="j6HrlzvIfZkLqVINV7sLyiSRfJE=">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</latexit>

du

<latexit sha1_base64="nnJkr/fMo0OvZyMNf3qApW8Ho3w=">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</latexit>

= 2rdr

=

Z

2⇡

0

2(R

2

)

32

3 d✓

<latexit sha1_base64="nbTtFxL9NQv78hM5LKPhftGO8To=">AAADL3icjVJNT9wwEH0E2gL92tIjl6irSvSyyoZK5VIJlUuPUHUBibArx+sFi2wSOQ7SKspf6X/of+iVXqteKo7lxk9gPBjUFlWto9jPb+aNZ8ZOy0xXNoq+zwXzC/fuP1hcWn746PGTp51nK7tVURupBrLICrOfikplOlcDq22m9kujxDTN1F56suXse6fKVLrIP9pZqQ6n4ijXEy2FJWrU2Xib6NyOmqgdNnFS6jaZGCGbeO3DMH41bK53620Tt21La5iE48QeKytGnW7Ui3iEd0Hfgy782C46l0gwRgGJGlMo5LCEMwhU9B2gjwglcYdoiDOENNsVWiyTtiYvRR6C2BOaj2h34Nmc9i5mxWpJp2T0G1KGeEmagvwMYXdayPaaIzv2b7Ebjulym9Ga+lhTYi2Oif2X7sbzf3WuFosJNrgGTTWVzLjqpI9Sc1dc5uEvVVmKUBLn8JjshrBk5U2fQ9ZUXLvrrWD7T/Z0rNtL71vjwmepcMpRZ7fZN3yHmuwl93JGyNLMt0RPov/nA7gLduNef70X7bzubr7zj2MRq3iBNXoBb7CJ99jGgLL5hC84w9fgc/At+BGcX7sGc17zHL+N4OIKDVOx2g==</latexit>

= 2R

3

3

2⇡

0

<latexit sha1_base64="9Mz8iZJwIQhbh62/edRMRgzMRdM=">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</latexit>

= 4⇡R

3

3

<latexit sha1_base64="fRg4M+b0JoPilJLHfmEAIuUHfUM=">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</latexit>

(14)

Exemple f (x, y ) = x

2

+ y

2

<latexit sha1_base64="3kJam02nqaMp8NhJdX+WULi/Lp4=">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</latexit>

x = y

<latexit sha1_base64="zyOmuutNNFgmwnGoOAn2c0PAUvA=">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</latexit>

x = 0

<latexit sha1_base64="Mph0riMHN7AtuCjQ2PoOV/+z/qE=">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</latexit>

La région comprise entre

cercle de rayon 4, cercle de rayon 3

3

<latexit sha1_base64="pOajbk9k7mKNCzzC3bye9YsLnrI=">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</latexit>

 r  4

4  r  ⇡ 2

<latexit sha1_base64="24gB7PE7sibV1EKf5R1LiUiDvq0=">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</latexit>

=

Z

2

4

Z

4

3

r

2

drd✓

<latexit sha1_base64="9cMzifGQalrGdnDkm0IOTEcwT6M=">AAADNHicjVJNSxxBEH1OPjTma6PHXIYsgZyW2XUhXgSJF48GXBUcd+np7dXG2Zmhp0dYhvk1+Q/5D171FMgtBPQi/gSryzaYSEh6mJ7Xr+rVVFVXUqS6tFH0bS549PjJ0/mFZ4vPX7x89br1ZmmnzCsj1UDmaW72ElGqVGdqYLVN1V5hlJgmqdpNjjecffdEmVLn2badFepgKg4zPdFSWKJGrbW1WGd2VMcTI2QdF7qp+00zvH/uNQ37rAz7YWiGvTgcm3Fsj5QVo1Y76kS8woeg60Ebfm3lrWvEGCOHRIUpFDJYwikESnr20UWEgrgD1MQZQprtCg0WSVuRlyIPQewx7Yd02vdsRmcXs2S1pL+k9BpShnhPmpz8DGH3t5DtFUd27N9i1xzT5Tajb+JjTYm1OCL2X7o7z//VuVosJljlGjTVVDDjqpM+SsVdcZmH96qyFKEgzuEx2Q1hycq7PoesKbl211vB9gv2dKw7S+9b4dJnqXDCUWe/sq/5DjXZC+7ljJClnW+JRqL75wA8BDu9TnelE33ut9c/+eFYwFu8wweagI9Yxya2MKBsvuAUZzgPvgbfgx/Bz1vXYM5rlvHbCq5uAD5htLU=</latexit>

Z

2

4

Z

4

3

x

2

+ y

2

rdrd✓

<latexit sha1_base64="hKPkGMP1VWR0f4z16qz6gz7P3KI=">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</latexit>

=

Z

2

4

r

3

3

4 3

!

d✓

<latexit sha1_base64="wwXtVYBywWE5VDpbl5p4zkWY/78=">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</latexit>

=

Z

2

4

64 27

3 d✓

<latexit sha1_base64="1Qr/RFQXIruIVjDHg7UYWSrhnZY=">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</latexit>

= 37✓

3

2

4

<latexit sha1_base64="gKkC7WG+rNuxbngRmN3kXM2nRUw=">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</latexit>

= 37⇡

12

<latexit sha1_base64="ANxhMbNjSGogieEMf0kfhhV78sw=">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</latexit>

= 37 3

⇣ ⇡ 2

⇡ 4

<latexit sha1_base64="Y0c0p6WL2UuA72scDBamMfUZurg=">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</latexit>

(15)

Exemple

La région comprise entre un cercle de rayon 1 et un de rayon 2.

Z

2⇡

0

Z

2

1

(2r cos ✓ r sin ✓ + 3)rdrd✓

<latexit sha1_base64="Y3oHsJ4umVMp/bsHzBMoh0TumtQ=">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</latexit>

f (x, y ) = 2x y + 3

<latexit sha1_base64="4ldsyiCX4msk6NqyfQRO3AMm99g=">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</latexit>

=

Z

2⇡

0

✓ 8 3

1 3

(2 cos ✓ sin ✓) + 3

✓ 4 2

1 2

d✓

<latexit sha1_base64="GNTdhoXL7Ngi7JHaRkuX4vo1dP4=">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</latexit>

= 7

3 (2 sin ✓ + cos ✓ ) + 9 2 ✓

2⇡

0

<latexit sha1_base64="PpxyCmeOKKkPhAMQNiBThsNKQHE=">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</latexit>

= 9⇡

<latexit sha1_base64="RQn8xJ/OE3jN24OE6VfxlnSutM4=">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</latexit>

=

Z

2⇡

0

Z

2

1

r

2

(2 cos ✓ sin ✓ ) + 3rdrd✓

<latexit sha1_base64="Vy77NSvUd6W6smV2b61Zc+TT/l4=">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</latexit>

(16)

Faites les exercices suivants

p. 857 # 1 à 14

p. 857 # 1 à 14

(17)

L’intégrale double permet aussi de calculer une aire.

A =

ZZ

R

1dA

<latexit sha1_base64="GwjiI/LaE1gElU4Hgt6P6XwiAuU=">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</latexit>

Car le volume d’un « cylindre » est l’aire de la base fois la hauteur.

(18)

On ne poussera pas trop dans cette direction, mais on peut regarder les fonctions polaires.

r = f (✓)

<latexit sha1_base64="1KqfFqnPS2Jqg3nWlBec6ENTKrU=">AAADAHicjVJNS8NAEH3Gr/pd9eglWAS9lFQFvQiiF48VrBZakSTd6mKahM1GKMWL/8GrXr2JV/+J/0Bv/gRnx634geiGbN6+mTeZmZ0gjWSmPe9pwBkcGh4ZLYyNT0xOTc8UZ+cOsyRXoaiFSZSoeuBnIpKxqGmpI1FPlfA7QSSOgvNdYz+6ECqTSXygu6k47vinsWzL0NdENdRWe7mpz4T2V06KJa/s8XJ/gooFJdhVTYqvaKKFBCFydCAQQxOO4COjp4EKPKTEHaNHnCIk2S5wiXHS5uQlyMMn9pz2Uzo1LBvT2cTMWB3SXyJ6FSldLJEmIT9F2PzNZXvOkQ37W+wexzS5dekb2FgdYjXOiP1L1/f8r87UotHGJtcgqaaUGVNdaKPk3BWTufupKk0RUuIMbpFdEQ5Z2e+zy5qMaze99dn+zJ6GNefQ+uZ4sVkKXHDU7kf2Pb5DSfaUe9klpGnnW6KRqHwfgJ/gcLVcWSt7++ul7R07HAUsYBHLNAEb2MYeqqjxHV3jBrfOlXPn3DsP767OgNXM48tyHt8AImCfAg==</latexit>

f

<latexit sha1_base64="PL3IzqRAS1r93IiHwtNX4hzzn6Q=">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</latexit>

(✓ ) = 2 cos(5✓ ) f (✓ ) = 1 + sin(✓ )

<latexit sha1_base64="mWQ23nwi3m9GLPSbO1OulH+EcOU=">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</latexit>

Rosace Cardioïde

(19)

Exemple

r

<latexit sha1_base64="gag/Ur/uJApKaHf/zx/opeful88=">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</latexit>

= ✓

r = ✓ 2

<latexit sha1_base64="wVVrpzVkl/zSNDmfjzus0mFbLl4=">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</latexit>

Z

2⇡

0

Z

2

1 rdrd✓

<latexit sha1_base64="XIgj32ISKsn0Y0Rz2EabBKJmN/4=">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</latexit>

=

Z

2⇡

0

r

2

2

2

d✓

<latexit sha1_base64="7Qk4kd/OxsdlF7HCEKOJzXKdzAI=">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</latexit>

=

Z

2⇡

0

2

2

2

8 d✓

<latexit sha1_base64="A2xLvi8NniELUl/PyeVRi3Eg+X8=">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</latexit>

= ✓

3

8

2⇡

0

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3

(20)

Faites les exercices suivants

p. 857 # 25 et 26

#26

Cette région

p. 857 # 23 et 24

Références

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