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3.4 INTÉGRALE TRIPLE

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Texte intégral

(1)

cours 21

3.4 INTÉGRALE

TRIPLE

(2)

Au dernier cours, nous avons vu

(3)

Au dernier cours, nous avons vu

Applications de l’intégrale double.

(4)

Au dernier cours, nous avons vu

Applications de l’intégrale double.

L’aire d’une surface.

(5)

Aujourd’hui, nous allons voir

(6)

Aujourd’hui, nous allons voir

Intégrale triple

(7)

Aujourd’hui, nous allons voir

Intégrale triple Type I, II et III

(8)

Aujourd’hui, nous allons voir

Intégrale triple Type I, II et III Volumes

(9)

Aujourd’hui, nous allons voir

Intégrale triple Type I, II et III Volumes

Applications

(10)

On sait déjà qu’il est difficile de visualiser les fonctions à plus de deux variables.

(11)

On sait déjà qu’il est difficile de visualiser les fonctions à plus de deux variables.

w = f (x, y, z)

(12)

On sait déjà qu’il est difficile de visualiser les fonctions à plus de deux variables.

w<latexit sha1_base64="BmTmZClXpLIB2bw14MWSaeMELpA=">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</latexit> = f (x, y, z)

Car son graphe est un sous-ensemble de . R<latexit sha1_base64="srBKwsXQ2HvP5UMdYEmRKcEubqE=">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</latexit> 4

(13)

On sait déjà qu’il est difficile de visualiser les fonctions à plus de deux variables.

w<latexit sha1_base64="BmTmZClXpLIB2bw14MWSaeMELpA=">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</latexit> = f (x, y, z)

Car son graphe est un sous-ensemble de . R<latexit sha1_base64="srBKwsXQ2HvP5UMdYEmRKcEubqE=">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</latexit> 4

Malgré ça, les idées développées pour l’intégrale double se généralisent très bien.

(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)

n,m,llim!1

Xn

i=1

Xm

j=1

Xl

k=1

f (xi, yj, zk) V =

ZZZ

B

f (x, y, z)dV

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(23)

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Xn

i=1

Xm

j=1

Xl

k=1

f (xi, yj, zk) V =

ZZZ

B

f (x, y, z)dV

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(Fubini)

(24)

n,m,llim!1

Xn

i=1

Xm

j=1

Xl

k=1

f (xi, yj, zk) V =

ZZZ

B

f (x, y, z)dV

<latexit sha1_base64="hueyAuUSxGkZ4lUwdTqsN9cBzEA=">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</latexit>

=

Z s

r

Z d

c

Z b

a

f (x, y, z)dxdydz

<latexit sha1_base64="wz+JAXh/2ioVmJWZjNSaoB82TWk=">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</latexit>

(Fubini)

(25)

Exemple

Z 3

2

Z 2

0

Z 2

1

x2yz dxdydz

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(26)

Exemple

Z 3

2

Z 2

0

Z 2

1

x2yz dxdydz

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=

Z 3

2

Z 2

0

x3yz 3

2

1dydz

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(27)

Exemple

Z 3

2

Z 2

0

Z 2

1

x2yz dxdydz

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=

Z 3

2

Z 2

0

x3yz 3

2

1dydz

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=

Z 3

2

Z 2

0

(23 13)yz

3 dydz

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(28)

Exemple

Z 3

2

Z 2

0

Z 2

1

x2yz dxdydz

<latexit sha1_base64="/KCFMB19Y6Ic5c0wlWH5RSNNJw8=">AAADHnicjVLLThRBFD20qIgPRl2y6Tgxupp0NyS4JLpxiYkDJAwz6UeBFXq6O93VhGbiJ/gP/oNb3bIjbvUPcMcncOpaEJUYrU5XnTr3nlv33qqkynVjguD7nHdj/uat2wt3Fu/eu/9gqffw0WZTtnWqhmmZl/V2Ejcq14UaGm1ytV3VKp4mudpKDl5Z+9ahqhtdFm9NV6ndabxf6D2dxobUpPdspAszicYrsgbjSNZwHPlH46g79kd+dpR12fGk1w8GgQz/Oggd6MONjbJ3jhEylEjRYgqFAoY4R4yG3w5CBKjI7WJGribSYld4j0VqW3opesRkDzjvc7fj2IJ7G7MRdcpTcv41lT6eUlPSrya2p/libyWyZf8WeyYxbW4d18TFmpI1eEf2X7pLz//V2VoM9vBCatCsqRLGVpe6KK10xWbu/1KVYYSKnMUZ7TVxKsrLPvuiaaR229tY7GfiaVm7T51vix8uS4VDidpdZT+TO9S0V9LLjshwllvikwj/fADXwWY0CFcG4ZvV/vpL9zgWsIwneM4XsIZ1vMYGhszmAz7hM754H70T79T7+tPVm3Oax/hteN8uAAiWqhg=</latexit>

=

Z 3

2

Z 2

0

x3yz 3

2

1dydz

<latexit sha1_base64="+37SIErM+yubxxxVlCE23PnWUpk=">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</latexit>

= 7 3

Z 3

2

Z 2

0

yzdydz

<latexit sha1_base64="QhwJaBym50zDJAp/fB0DB2rl03k=">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</latexit>

=

Z 3

2

Z 2

0

(23 13)yz

3 dydz

<latexit sha1_base64="kZ2Iqi56eOTJhlWp4/Vh9Be98N0=">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</latexit>

(29)

Exemple

Z 3

2

Z 2

0

Z 2

1

x2yz dxdydz

<latexit sha1_base64="/KCFMB19Y6Ic5c0wlWH5RSNNJw8=">AAADHnicjVLLThRBFD20qIgPRl2y6Tgxupp0NyS4JLpxiYkDJAwz6UeBFXq6O93VhGbiJ/gP/oNb3bIjbvUPcMcncOpaEJUYrU5XnTr3nlv33qqkynVjguD7nHdj/uat2wt3Fu/eu/9gqffw0WZTtnWqhmmZl/V2Ejcq14UaGm1ytV3VKp4mudpKDl5Z+9ahqhtdFm9NV6ndabxf6D2dxobUpPdspAszicYrsgbjSNZwHPlH46g79kd+dpR12fGk1w8GgQz/Oggd6MONjbJ3jhEylEjRYgqFAoY4R4yG3w5CBKjI7WJGribSYld4j0VqW3opesRkDzjvc7fj2IJ7G7MRdcpTcv41lT6eUlPSrya2p/libyWyZf8WeyYxbW4d18TFmpI1eEf2X7pLz//V2VoM9vBCatCsqRLGVpe6KK10xWbu/1KVYYSKnMUZ7TVxKsrLPvuiaaR229tY7GfiaVm7T51vix8uS4VDidpdZT+TO9S0V9LLjshwllvikwj/fADXwWY0CFcG4ZvV/vpL9zgWsIwneM4XsIZ1vMYGhszmAz7hM754H70T79T7+tPVm3Oax/hteN8uAAiWqhg=</latexit>

=

Z 3

2

Z 2

0

x3yz 3

2

1dydz

<latexit sha1_base64="+37SIErM+yubxxxVlCE23PnWUpk=">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</latexit>

= 7 3

Z 3

2

Z 2

0

yzdydz

<latexit sha1_base64="QhwJaBym50zDJAp/fB0DB2rl03k=">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</latexit>

= 7 3

Z 3

2

y2z 2

2

0dz

<latexit sha1_base64="CcRTGlIQvvlHBihbeDGyd2+dC/s=">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</latexit>

=

Z 3

2

Z 2

0

(23 13)yz

3 dydz

<latexit sha1_base64="kZ2Iqi56eOTJhlWp4/Vh9Be98N0=">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</latexit>

(30)

Exemple

Z 3

2

Z 2

0

Z 2

1

x2yz dxdydz

<latexit sha1_base64="/KCFMB19Y6Ic5c0wlWH5RSNNJw8=">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</latexit>

=

Z 3

2

Z 2

0

x3yz 3

2

1dydz

<latexit sha1_base64="+37SIErM+yubxxxVlCE23PnWUpk=">AAADUHicjVLLTttAFD1xSkvTB4Euu7EaVeoqspNKZVMJ0U2XIDWAREhkTybpCMe2xmNU4+Yv+jXdwhew4w/aP2AHdy6DxEN9jGXPmXPPuTP3euI8UYUJgvOG13y09PjJ8tPWs+cvXq60V9d2iqzUQg5ElmR6L44KmahUDowyidzLtYzmcSJ348NPNr57JHWhsvSLqXJ5MI9mqZoqERmixu3ux6FKzbg36vMcjHrDqY5E/W3Ur44XdX8x3FSz7+Nw1PMn1eR43O4E3YCH/xCEDnTgxla22ljCEBNkECgxh0QKQzhBhIKefYQIkBN3gJo4TUhxXGKBFnlLUklSRMQe0ndGq33HprS2OQt2C9oloVeT08db8mSk04Ttbj7HS85s2T/lrjmnPVtFc+xyzYk1+Ersv3w3yv/12VoMpljnGhTVlDNjqxMuS8ldsSf3b1VlKENOnMUTimvCgp03ffbZU3DttrcRx3+x0rJ2LZy2xO+/VmcdR24HeyqrlMxo7tR1nTX/bUXanLteETL05f9Jtye8f1cegp1eN+x3w+33nY1Nd4+W8Rpv8I7uygds4DO2MKBz/MBPnODUO/MuvMtm41rquRmvcGc0W1dqCLOT</latexit>

= 7 3

Z 3

2

Z 2

0

yzdydz

<latexit sha1_base64="QhwJaBym50zDJAp/fB0DB2rl03k=">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</latexit>

= 7 3

Z 3

2

y2z 2

2

0dz

<latexit sha1_base64="CcRTGlIQvvlHBihbeDGyd2+dC/s=">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</latexit>

=

Z 3

2

Z 2

0

(23 13)yz

3 dydz

<latexit sha1_base64="kZ2Iqi56eOTJhlWp4/Vh9Be98N0=">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</latexit>

= 7 3

Z 3

2

(22 02)z

2 dz

<latexit sha1_base64="D2pTsJEATJNTgISnXv93WKcKMpg=">AAADTXicjVJdT9RAFD1bRBC/Fnz0pWFjAjFu2q4JvpgQfPERExZIWHbTzs7ihG7bTKckS9Mfwa/xVX6Bz/4D+QG8GeOdu0OiEpVp2p4595w7c+9MUqSqNEHwteUt3Fu8v7T8YOXho8dPnrZX1/bLvNJC9kWe5vowiUuZqkz2jTKpPCy0jKdJKg+S03c2fnAmdanybM/MCnk8jU8yNVEiNkSN2i/fDiY6FvVWU/eagcrMKBr25tRGNIxeBcNo87ypo8Yfn4/anaAb8PBvg9CBDtzYzVdbixhgjBwCFaaQyGAIp4hR0nOEEAEK4o5RE6cJKY5LNFghb0UqSYqY2FP6ntDsyLEZzW3Okt2CVknp1eT08YI8Oek0Ybuaz/GKM1v2b7lrzmn3NqN/4nJNiTX4SOz/fDfKu/psLQYTvOEaFNVUMGOrEy5LxV2xO/d/qcpQhoI4i8cU14QFO2/67LOn5Nptb2OOf2OlZe1cOG2Fq39WZx1nbgW7K6uUzGju1LzOmk9bkbbgrs8IGfryedLtCf+8K7fBftQNe93ww+vO9o67R8t4jnVs0F3ZwjbeYxd92scFPuEzLr0v3rX33fsxl3ot53mG38bC0k/G27KW</latexit>

(31)

Exemple

Z 3

2

Z 2

0

Z 2

1

x2yz dxdydz

<latexit sha1_base64="/KCFMB19Y6Ic5c0wlWH5RSNNJw8=">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</latexit>

=

Z 3

2

Z 2

0

x3yz 3

2

1dydz

<latexit sha1_base64="+37SIErM+yubxxxVlCE23PnWUpk=">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</latexit>

= 7 3

Z 3

2

Z 2

0

yzdydz

<latexit sha1_base64="QhwJaBym50zDJAp/fB0DB2rl03k=">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</latexit>

= 7 3

Z 3

2

y2z 2

2

0dz

<latexit sha1_base64="CcRTGlIQvvlHBihbeDGyd2+dC/s=">AAADUHicjVLLbtNAFD1xWigpjxSWbCwipK4i20EqG6SqbFgWqWkrNU1kTyZhVMe2xuNKqfFf8DVs4QvY9Q/KH7Ard26mErTiMZbtM+eec2funUmKVJUmCC5bXntt/d79jQedzYePHj/pbj09LPNKCzkUeZrr4yQuZaoyOTTKpPK40DJeJKk8Ss7e2vjRudSlyrMDsyzk6SKeZ2qmRGyImnT7b0YzHYt6p6kHzUhlZhKNBytqOY4umjpqRntq/nESjCN/ejHp9oJ+wMO/C0IHenBjP99qrWOEKXIIVFhAIoMhnCJGSc8JQgQoiDtFTZwmpDgu0aBD3opUkhQxsWf0ndPsxLEZzW3Okt2CVknp1eT08ZI8Oek0Ybuaz/GKM1v2T7lrzmn3tqR/4nItiDX4QOy/fDfK//XZWgxmeM01KKqpYMZWJ1yWirtid+7/UpWhDAVxFk8prgkLdt702WdPybXb3sYcv2KlZe1cOG2F73+tzjrO3Qp2V1YpmdHcqVWdNZ+2Im3BXV8SMvTl86TbE96+K3fBYdQPB/3w/ave7p67Rxt4jhfYpruyg128wz6GtI9P+Iwv+Op983541+3WSuq5P57ht9Hu/AQG6LPH</latexit>

= 14 3

Z 3

2

zdz

<latexit sha1_base64="ZmGX02yWfyZApit2u56mQrmQpIA=">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</latexit>

=

Z 3

2

Z 2

0

(23 13)yz

3 dydz

<latexit sha1_base64="kZ2Iqi56eOTJhlWp4/Vh9Be98N0=">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</latexit>

= 7 3

Z 3

2

(22 02)z

2 dz

<latexit sha1_base64="D2pTsJEATJNTgISnXv93WKcKMpg=">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</latexit>

(32)

Exemple

Z 3

2

Z 2

0

Z 2

1

x2yz dxdydz

<latexit sha1_base64="/KCFMB19Y6Ic5c0wlWH5RSNNJw8=">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</latexit>

=

Z 3

2

Z 2

0

x3yz 3

2

1dydz

<latexit sha1_base64="+37SIErM+yubxxxVlCE23PnWUpk=">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</latexit>

= 7 3

Z 3

2

Z 2

0

yzdydz

<latexit sha1_base64="QhwJaBym50zDJAp/fB0DB2rl03k=">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</latexit>

= 7 3

Z 3

2

y2z 2

2

0dz

<latexit sha1_base64="CcRTGlIQvvlHBihbeDGyd2+dC/s=">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</latexit>

= 14 3

Z 3

2

zdz

<latexit sha1_base64="ZmGX02yWfyZApit2u56mQrmQpIA=">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</latexit>

= 14 3

✓ 9 2

4 2

=

Z 3

2

Z 2

0

(23 13)yz

3 dydz

<latexit sha1_base64="kZ2Iqi56eOTJhlWp4/Vh9Be98N0=">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</latexit>

= 7 3

Z 3

2

(22 02)z

2 dz

<latexit sha1_base64="D2pTsJEATJNTgISnXv93WKcKMpg=">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</latexit>

(33)

Exemple

Z 3

2

Z 2

0

Z 2

1

x2yz dxdydz

<latexit sha1_base64="/KCFMB19Y6Ic5c0wlWH5RSNNJw8=">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</latexit>

=

Z 3

2

Z 2

0

x3yz 3

2

1dydz

<latexit sha1_base64="+37SIErM+yubxxxVlCE23PnWUpk=">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</latexit>

= 7 3

Z 3

2

Z 2

0

yzdydz

<latexit sha1_base64="QhwJaBym50zDJAp/fB0DB2rl03k=">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</latexit>

= 7 3

Z 3

2

y2z 2

2

0dz

<latexit sha1_base64="CcRTGlIQvvlHBihbeDGyd2+dC/s=">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</latexit>

= 14 3

Z 3

2

zdz

<latexit sha1_base64="ZmGX02yWfyZApit2u56mQrmQpIA=">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</latexit>

= 14 3

✓ 9 2

4 2

<latexit sha1_base64="0PzgrBorf3QRdFYJM4eyi8MpNi4=">AAADVXicjVJdS+NAFD1t1VX3w6qPvgTLgvuwJbEF1wdB3BcfFbYqWJFknNbBNAmTiVBCfoi/xlf9BeI/WH+E4J3bEVbF3Z2Q5My559yZe2eiLFa58f37Wr0xNT3zYXZu/uOnz18WmotLB3laaCF7Io1TfRSFuYxVIntGmVgeZVqGoyiWh9HFTxs/vJQ6V2nyy4wzeTIKh4kaKBEaok6bna3+QIeiDLpV2an6sRyYNW9CbVblevV9grsW97Uanptv3mmz5bd9Ht5bEDjQght76WJtGn2cIYVAgREkEhjCMULk9BwjgI+MuBOUxGlCiuMSFebJW5BKkiIk9oK+Q5odOzahuc2Zs1vQKjG9mpwevpInJZ0mbFfzOF5wZsu+l7vknHZvY/pHLteIWINzYv/le1b+r8/WYjDAD65BUU0ZM7Y64bIU3BW7c++PqgxlyIiz+IzimrBg53OfPfbkXLvtbcjx36y0rJ0Lpy3w8NfqrOPSrWB3ZZWSGc2dmtRZ8mkr0mbc9TEhQ18+T7o9weu78hYcrLeDTjvY77a2d9w9msUKVrFGd2UD29jFHnq0jytc4wa39bv6Y2OqMTOR1mvOs4wXo7HwBGhctJs=</latexit>

= 35 3

<latexit sha1_base64="BP/eBOxtYtYmzvE6JZyBJMxZk+I=">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</latexit>

=

Z 3

2

Z 2

0

(23 13)yz

3 dydz

<latexit sha1_base64="kZ2Iqi56eOTJhlWp4/Vh9Be98N0=">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</latexit>

= 7 3

Z 3

2

(22 02)z

2 dz

<latexit sha1_base64="D2pTsJEATJNTgISnXv93WKcKMpg=">AAADTXicjVJdT9RAFD1bRBC/Fnz0pWFjAjFu2q4JvpgQfPERExZIWHbTzs7ihG7bTKckS9Mfwa/xVX6Bz/4D+QG8GeOdu0OiEpVp2p4595w7c+9MUqSqNEHwteUt3Fu8v7T8YOXho8dPnrZX1/bLvNJC9kWe5vowiUuZqkz2jTKpPCy0jKdJKg+S03c2fnAmdanybM/MCnk8jU8yNVEiNkSN2i/fDiY6FvVWU/eagcrMKBr25tRGNIxeBcNo87ypo8Yfn4/anaAb8PBvg9CBDtzYzVdbixhgjBwCFaaQyGAIp4hR0nOEEAEK4o5RE6cJKY5LNFghb0UqSYqY2FP6ntDsyLEZzW3Okt2CVknp1eT08YI8Oek0Ybuaz/GKM1v2b7lrzmn3NqN/4nJNiTX4SOz/fDfKu/psLQYTvOEaFNVUMGOrEy5LxV2xO/d/qcpQhoI4i8cU14QFO2/67LOn5Nptb2OOf2OlZe1cOG2Fq39WZx1nbgW7K6uUzGju1LzOmk9bkbbgrs8IGfryedLtCf+8K7fBftQNe93ww+vO9o67R8t4jnVs0F3ZwjbeYxd92scFPuEzLr0v3rX33fsxl3ot53mG38bC0k/G27KW</latexit>

(34)

Exemple

ZZZ

x2+y2+z2R

3 + x3 + 4 sin z dV

<latexit sha1_base64="QekqMzSTuhn/UIxIJI9F1m6SZqQ=">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</latexit>

(35)

Exemple

ZZZ

x2+y2+z2R

3 + x3 + 4 sin z dV

<latexit sha1_base64="QekqMzSTuhn/UIxIJI9F1m6SZqQ=">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</latexit>

=

ZZZ

x2+y2+z2R

3 dV +

ZZZ

x2+y2+z2R

x3 dV +

ZZZ

x2+y2+z2R

4 sin z dV

<latexit sha1_base64="MPjw5JZBrTDUJEAYVw0RriD1yPU=">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</latexit>

(36)

3 fois le volume de la sphère

Exemple

ZZZ

x2+y2+z2R

3 + x3 + 4 sin z dV

<latexit sha1_base64="QekqMzSTuhn/UIxIJI9F1m6SZqQ=">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</latexit>

=

ZZZ

x2+y2+z2R

3 dV +

ZZZ

x2+y2+z2R

x3 dV +

ZZZ

x2+y2+z2R

4 sin z dV

<latexit sha1_base64="MPjw5JZBrTDUJEAYVw0RriD1yPU=">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</latexit>

(37)

0 car fonction impaire.

3 fois le volume de la sphère

Exemple

ZZZ

x2+y2+z2R

3 + x3 + 4 sin z dV

<latexit sha1_base64="QekqMzSTuhn/UIxIJI9F1m6SZqQ=">AAADTXicjVLLbtNAFD1xKS3l0RSWbCwiJKRIkZ1UgmVFNywLImmluolsZ1pGdWxjj6umUT+iX8MWvoA1fwAfwA4hztxOJaDiMZbtM+eec2funUnKTNcmCD61vKUbyzdXVm+t3b5z9956e+P+qC6aKlXDtMiKai+Ja5XpXA2NNpnaKysVz5JM7SbH2za+e6KqWhf5azMv1cEsPsr1oU5jQ2rS7kZa69xMFqfjfnfO92zcjzL11n91PuiejgfdzajWuX8W+dPRpN0JeoEM/zoIHejAjZ1io7WMCFMUSNFgBoUchjhDjJrPPkIEKMkdYEGuItISVzjHGr0NVYqKmOwxv0ec7Ts259zmrMWdcpWMb0Wnj8f0FNRVxHY1X+KNZLbsn3IvJKfd25z/xOWakTV4Q/Zfvivl//psLQaHeCY1aNZUCmOrS12WRrpid+7/VJVhhpKcxVPGK+JUnFd99sVTS+22t7HEP4vSsnaeOm2DL3+tzjpO3Ap2V1aphKmkU5d1LuS0NbWldH1OZPiV8+TtCX+/K9fBqN8LB73w5WZn67m7R6t4iEd4wrvyFFt4gR0MuY8LvMN7fPA+el+9b973S6nXcp4H+GUsrfwAw82yOQ==</latexit>

=

ZZZ

x2+y2+z2R

3 dV +

ZZZ

x2+y2+z2R

x3 dV +

ZZZ

x2+y2+z2R

4 sin z dV

<latexit sha1_base64="MPjw5JZBrTDUJEAYVw0RriD1yPU=">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</latexit>

(38)

0 car fonction impaire.

3 fois le volume de la sphère

Exemple

ZZZ

x2+y2+z2R

3 + x3 + 4 sin z dV

<latexit sha1_base64="QekqMzSTuhn/UIxIJI9F1m6SZqQ=">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</latexit>

=

ZZZ

x2+y2+z2R

3 dV +

ZZZ

x2+y2+z2R

x3 dV +

ZZZ

x2+y2+z2R

4 sin z dV

<latexit sha1_base64="MPjw5JZBrTDUJEAYVw0RriD1yPU=">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</latexit>

= 4⇡<latexit sha1_base64="s4PV1I2U262MV1tbYyp9F3qaeb8=">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</latexit> R3

(39)

Faites les exercices suivants

p. 880 # 1 et 2 p. 880 # 1 et 2

(40)

Type I

(41)

a<latexit sha1_base64="KlSGEXQz4iKFrU+Hg9x+KbTHwjw=">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</latexit>  x  b Type I

(42)

a<latexit sha1_base64="KlSGEXQz4iKFrU+Hg9x+KbTHwjw=">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</latexit>  x  b

f<latexit sha1_base64="uvKZu3tUHJNBHfYrjKYhB2+wQu8=">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</latexit> 1(x)  y  f2(x) Type I

(43)

a<latexit sha1_base64="KlSGEXQz4iKFrU+Hg9x+KbTHwjw=">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</latexit>  x  b

f<latexit sha1_base64="uvKZu3tUHJNBHfYrjKYhB2+wQu8=">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</latexit> 1(x)  y  f2(x) g<latexit sha1_base64="Pi7gpo/nt3bx7Af4ww73gqrY+R8=">AAADEnicjVLBTtVAFD0UBESEpy7ZNL6YQGJeWiDBJcENS0x8QALkpS3Dc0JfW9opsb6w4B/4B7awZWfc+gP+ge74BM5cB6IQotN0eubce27vvXPjItWVCYIfI97o2JPxicmnU8+mn8/Mtl683KzyukxUN8nTvNyOo0qlOlNdo02qtotSRYM4VVvx4Xtr3zpWZaXz7KNpCrU3iPqZPtBJZEj1WnP9Xjj/+W2zsJuqI/+L7P3eolC9VjvoBLL8hyB0oA23NvLWNXaxjxwJagygkMEQp4hQ8dlBiAAFuT0MyZVEWuwKJ5iitqaXokdE9pB7n6cdx2Y825iVqBP+JeVbUunjDTU5/Upi+zdf7LVEtuxjsYcS0+bW8Bu7WAOyBp/I/kt36/m/OluLwQHeSQ2aNRXC2OoSF6WWrtjM/T+qMoxQkLN4n/aSOBHlbZ990VRSu+1tJPaf4mlZe06cb41fLkuFY4na3GU/lDvUtBfSy4bIcJdb4kiE9wfgIdhc7IRLnfDDcnt1zQ3HJObwGvOcgBWsYh0b6DKbU5zjApfemXflffW+/Xb1RpzmFf5a3vcbHeSkvQ==</latexit> 1(x, y)  z  g2(x, y)

Type I

(44)

a<latexit sha1_base64="KlSGEXQz4iKFrU+Hg9x+KbTHwjw=">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</latexit>  x  b

f1(x)  y  f2(x)

<latexit sha1_base64="uvKZu3tUHJNBHfYrjKYhB2+wQu8=">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</latexit>

Type I

g<latexit sha1_base64="Pi7gpo/nt3bx7Af4ww73gqrY+R8=">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</latexit> 1(x, y)  z  g2(x, y)

(45)

a<latexit sha1_base64="KlSGEXQz4iKFrU+Hg9x+KbTHwjw=">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</latexit>  x  b

f1(x)  y  f2(x)

<latexit sha1_base64="uvKZu3tUHJNBHfYrjKYhB2+wQu8=">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</latexit>

Type I

c<latexit sha1_base64="OIjcidnsYy9HU3PdLcU0jpODZC4=">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</latexit>  y  d

g<latexit sha1_base64="Pi7gpo/nt3bx7Af4ww73gqrY+R8=">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</latexit> 1(x, y)  z  g2(x, y)

(46)

a<latexit sha1_base64="KlSGEXQz4iKFrU+Hg9x+KbTHwjw=">AAADAnicjVJNT9tAEH0xtKT0K5RjL1ajSj1FdotUjhFcOKYSSZAgqmyz0BWObex1RBRx4z9whWtvqNf+kf4DuPETeDvdVC2oatfy+u2beeOZ2YmLVFcmCH40vIXFR4+Xmk+Wnz57/uJla+XVoMrrMlH9JE/zcieOKpXqTPWNNqnaKUoVjeNUDeOjTWsfTlRZ6TzbNtNCjcbRYaYPdBIZUqNoL1XH/ons8edWO+gEsvyHIHSgDbd6eesWe9hHjgQ1xlDIYIhTRKj47CJEgILcCDNyJZEWu8Iplqmt6aXoEZE94n7I065jM55tzErUCf+S8i2p9PGWmpx+JbH9my/2WiJb9m+xZxLT5jblN3axxmQNvpD9l27u+b86W4vBAdalBs2aCmFsdYmLUktXbOb+b1UZRijIWbxPe0mciHLeZ180ldRuexuJ/Vo8LWvPifOtceOyVJhI1Omv7Gdyh5r2Qno5JTLc5ZY4EuH9AXgIBu874YdO+Gmt3d1ww9HEa7zBO07AR3SxhR76zOYY57jApXfmffWuvG8/Xb2G06zij+V9vwO10Z/2</latexit>  x  b

f1(x)  y  f2(x)

<latexit sha1_base64="uvKZu3tUHJNBHfYrjKYhB2+wQu8=">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</latexit>

Type I

c<latexit sha1_base64="OIjcidnsYy9HU3PdLcU0jpODZC4=">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</latexit>  y  d

f1(y)  x  f2(y)

<latexit sha1_base64="VKRXxE1WJMuIqrPzM5iKHs7PkI8=">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</latexit>

g<latexit sha1_base64="Pi7gpo/nt3bx7Af4ww73gqrY+R8=">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</latexit> 1(x, y)  z  g2(x, y)

Références

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