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3.1 INTÉGRALE DOUBLE

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Texte intégral

(1)

cours 17

3.1 INTÉGRALE

DOUBLE

(2)

Aujourd’hui, nous allons voir

Double somme de Riemann.

Intégrale partielle.

Intégrale itérée.

(3)

nlim!1

Xn

i=1

f (xi ) xi

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=

Z b

a

f (x) dx

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Sommes de Riemann

(4)

nlim!1 m!1

Xn

i=1

Xm

k=1

f (xi , yk) Aik

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=

ZZ

R

(x, y)dA

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L’intégrale double

(5)

On peut déduire certaines propriétés de l’intégrale double définie

Aire(R) = 0

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Si

ZZ

R

f (x, y)dA = 0

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alors 1)

ZZ

R

1dA = Aire(R)

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2)

ZZ

R

(kf (x, y) + rg(xy))dA = k

ZZ

R

f (x, y)dA + r

ZZ

R

g(xy)dA

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3) Linéarité

(6)

R<latexit sha1_base64="qSgwlcn2bjS52UBt57Iu1ni6ZTQ=">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</latexit> = R1 [ R2 [ · · · [ Rn

=

[n

k=1

Rk

<latexit sha1_base64="9S/bb1pN803WtJS+C3x0izryzts=">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</latexit>

8i 6= j

<latexit sha1_base64="T1X+9XNAnZJVhka4z+pMsZA/L0k=">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</latexit> Ri \ Rj = ;<latexit sha1_base64="I4St4XeBUh8ta5ajgWPekn759Oo=">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</latexit>

ZZ

R

f (x, y)dA =

Xn

k=1

ZZ

Rk

f (x, y)dA

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3) Décomposition de la région

(7)

Bien que cette généralisation est théoriquement intéressante, d’un point de vue calculatoire, elle ne nous est pas d’une très grande aide.

On aimerait pouvoir utiliser le théorème fondamental du calcul et utiliser les primitives pour calculer les intégrales indéfinies.

Mais des primitives par rapport à quoi?

(8)

Une fonction qui dépend seulement de x<latexit sha1_base64="HA6BPTQP3UBB+d9iHJHQ4r2LiZk=">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</latexit>

Une fonction qui dépend seulement de y<latexit sha1_base64="ADnGsQtcomAbrlTQt0snNnnV3IQ=">AAAC9nicjVJNS8NAEH3Gr/pd9eglWARPJVVBj0UvHluwVahFkritS9MkJJtiKP4Cr3r1Jl79O/4DvfkTnB1XUYvohmzevpk3mZkdLw5kqhznacwan5icmi7MzM7NLywuFZdXmmmUJb5o+FEQJSeem4pAhqKhpArESZwIt+8F4tjrHWj78UAkqYzCI5XHot13u6HsSN9VRNXzs2LJKTu87FFQMaAEs2pR8RWnOEcEHxn6EAihCAdwkdLTQgUOYuLaGBKXEJJsF7jCLGkz8hLk4RLbo71Lp5ZhQzrrmCmrffpLQG9CShsbpInILyGs/2azPePImv0t9pBj6txy+nomVp9YhQti/9J9eP5Xp2tR6GCPa5BUU8yMrs43UTLuis7c/lKVoggxcRqfkz0h7LPyo882a1KuXffWZfsze2pWn33jm+HFZCkw4Kj5Z/ZDvkNJ9ph7mRNStPMt0UhUfg7AKGhulSvbZae+U6rum+EoYA3r2KQJ2EUVh6ihwdlc4wa31qV1Z91bD++u1pjRrOLbsh7fAOhamz8=</latexit>

Exemple

De la même manière qu’on a fait des dérivée partielle en gardant constant une ou des variables, on peut intégrer partiellement en

conservant les autres variable constante.

Z

xy + y2dx

<latexit sha1_base64="StDbpUw4mzChjCJSKE/soyjSQZs=">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</latexit>

= xy2

2 + y2x + C(y)

<latexit sha1_base64="kTUJMbsUAvIDbkzPAcldnKys678=">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</latexit>

Z

xy + y2dy

<latexit sha1_base64="F2ZPWV3L5GLg40pyjXcBXPSbxIA=">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</latexit>

= xy2

2 + y3

3 + C(x)

<latexit sha1_base64="gYgzaSQuqlH8N0eO+vVQHdl+OUw=">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</latexit>

(9)

Faites les exercices suivants

p.843 # 1 et 2

p.843 # 1 et 2

(10)

Regardons comment on peut utiliser l’intégration par rapport à une variable pour calculer des intégrales doubles.

(11)

Z b a

f (x, c)dx

!

y1

<latexit sha1_base64="kFGWbUz4XxYc8OniJR4zSS1s+KU=">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</latexit>

Z d c

f (a, y)dy

!

x1

<latexit sha1_base64="lfBAmeFh6W4nr3LIMOdr2lDDZEE=">AAADH3icjVLNbtQwEP6a0tIfKEs5colYIW0ltEoAiR4r4MCxSGxbqSkrx+vdWvUmkeNUjVa8Au/AO3CFKzfEtW9AbzwC48FFtFVFHcX5/M18k5nx5JXRtUuS07lo/tbC4u2l5ZXVO3fX7nXur+/UZWOlGsjSlHYvF7UyulADp51Re5VVYpobtZsfvfL23WNla10W71xbqYOpmBR6rKVwRA07vcyosetlunBD+X407okn7UY8ajOrJ4duI3utjBPxyTAddrpJP+EVXwVpAF2EtV12fiHDCCUkGkyhUMARNhCo6dlHigQVcQeYEWcJabYrfMAKaRvyUuQhiD2ifUKn/cAWdPYxa1ZL+ouh15IyxmPSlORnCfu/xWxvOLJnr4s945g+t5a+eYg1JdbhkNj/6c49b6rztTiMsck1aKqpYsZXJ0OUhrviM4//qcpRhIo4j0dkt4QlK8/7HLOm5tp9bwXbf7KnZ/1ZBt8GZyFLhWOO2v7NfsZ3qMlecS9bQo52viUaifTyAFwFO0/76bN+8vZ5d+tlGI4lPMQj9GgCXmALb7CNAWXzEZ/xBV+jT9G36Hv0449rNBc0D3BhRae/AZGHqqI=</latexit>

nX1

k=0

Z d c

f (a + k xk, y)dy

!

xk

<latexit sha1_base64="6RHZbQxw2F0LZiMfsZHJNN38G9M=">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</latexit>

nX1

k=0

Z b a

f (x, c + k yk)dx

!

yk

<latexit sha1_base64="GUVNG667Ac754mXthY+S/E1Uiss=">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</latexit>

(12)

nX1

k=0

Z d c

f (a + k xk, y)dy

!

xk

<latexit sha1_base64="6RHZbQxw2F0LZiMfsZHJNN38G9M=">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</latexit>

nX1

k=0

Z b a

f (x, c + k yk)dx

!

yk

<latexit sha1_base64="GUVNG667Ac754mXthY+S/E1Uiss=">AAADO3icjVJLb9QwEP4aXqW8FjhyiVghbQWsEqhULkgV9MCxSGxbqWkjx+vdWnEecpyqUbR/iP/Af+BarlwAceDCnbFxeVUIHMX5/M18k5nxZLWSjYmik6Xg3PkLFy8tX165cvXa9RuDm7e2m6rVXEx4pSq9m7FGKFmKiZFGid1aC1ZkSuxk+XNr3zkSupFV+cp0tdgv2LyUM8mZISodbCZNW6R9/jRaHPTlw3iRKDEzo0SWJmUH2Wx0/IDfz5NNoQwLuzRfDafHiZbzQ7P6k0wHw2gcuRWeBbEHQ/i1VQ2+IMEUFThaFBAoYQgrMDT07CFGhJq4ffTEaULS2QUWWCFtS16CPBixOe1zOu15tqSzjdk4Nae/KHo1KUPcI01Ffpqw/Vvo7K2LbNm/xe5dTJtbR9/MxyqINTgk9l+6U8//1dlaDGZ44mqQVFPtGFsd91Fa1xWbefhLVYYi1MRZPCW7Jsyd8rTPodM0rnbbW+bsH5ynZe2Ze98WH32WAkcuavcj+97doSR77XrZETK0u1uikYj/HICzYPvROH48jl6uDTee+eFYxh3cxYgmYB0beIEtTCib13iLE7wL3gTvg0/B5++uwZLX3MZvK/j6DdritjI=</latexit>

nlim!1

<latexit sha1_base64="hrsudP0wu+jVExR6j77Bcuul6nE=">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</latexit>

n<latexit sha1_base64="hrsudP0wu+jVExR6j77Bcuul6nE=">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</latexit> lim!1

=

Z d

c

Z b a

f (x, y)dx

!

dy

<latexit sha1_base64="HM3BzEMf7aCpCktWt41HKqr49/4=">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</latexit>

=

Z b

a

Z d c

f (x, y)dy

!

dx

<latexit sha1_base64="Kvr2XxoLGX0ZH25mSsxW9LMxNDU=">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</latexit>

=

Z b

a

Z d

c

f (x, y)dydx

<latexit sha1_base64="7nBR0adggOXS35WFZnfnBJjqppA=">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</latexit>

=

Z d

c

Z b

a

f (x, y)dxdy

<latexit sha1_base64="+HAvDRBjGnQPL0LNznCPlAbXd4I=">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</latexit>

On peut laisser tomber les parenthèses

(13)

Théorème de Fubini:

ZZ

R

f (x, y)dA =

Z b

a

Z d

c

f (x, y)dydx =

Z d

c

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<latexit sha1_base64="MikuozMpT8jJwOQ9F0x1eQ8RaoA=">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</latexit>

R<latexit sha1_base64="Z04kHP/TLKYzxYap5jL7KJjoSV0=">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</latexit> = [a, b] ⇥ [c, d] =<latexit sha1_base64="ynaYvFLnk1ZQmGVE/mpHfGEEEbg=">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</latexit> {(x, y)|a  x  b, c  y  d} Si f<latexit sha1_base64="yaOGmVow1D/1W4I/oozk9nSKCAA=">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</latexit> (x, y) est continue sur

alors

(14)

Exemple

Z 1

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<latexit sha1_base64="BK8P7BYNX+5IIbr586SkkQZp0ss=">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</latexit>

=

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<latexit sha1_base64="Gu1gXW1HCULV6qyEDobS6VabyOg=">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</latexit>

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<latexit sha1_base64="8Nn2ov5Qyl1skV/HoTWBYBP8cV8=">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</latexit>

=

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<latexit sha1_base64="fBtKCsMQexKWoVJbE52e1ktCBaY=">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</latexit>

=

Z 1

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<latexit sha1_base64="g9HxAj9eyhUnuK36tCIImD7HvVc=">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</latexit>

= 3y2 4

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<latexit sha1_base64="pRl/RJemIIg4yEP8O3W4xAhNtvk=">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</latexit>

= 3 4

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=

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=

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<latexit sha1_base64="K+ownmLKAvCOlGhKWRmfpldBOZE=">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</latexit>

= x2 4

2

1

<latexit sha1_base64="o18BEkIZs+wLQWbFXXC6ehBstd0=">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</latexit>

= 4 4

1 4

<latexit sha1_base64="rh5w5MEG34GvrG5jsFZIG7o+plo=">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</latexit>

= 3 4

<latexit sha1_base64="+/eKNL6c9jhp/3NyjFOnCwgdPOo=">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</latexit>

Z 1

0

Z 2

1

xy dxdy

<latexit sha1_base64="BK8P7BYNX+5IIbr586SkkQZp0ss=">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</latexit>

(15)

Faites les exercices suivants

p.843 # 3 à 6

p.843 # 3 à 6

(16)

Ici est constanty<latexit sha1_base64="ADnGsQtcomAbrlTQt0snNnnV3IQ=">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</latexit>

Exemple

Z 2

0

Z 3

0

px + y dxdy

<latexit sha1_base64="7pP4Gnp/ANy+NKf3xY5PQw6EZZ8=">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</latexit>

u<latexit sha1_base64="+vDaOsQSJmFbvNQ9staFQHwLzd4=">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</latexit> = x + y du<latexit sha1_base64="gNvm+ShnHnrF7be0i1nE2d5HyAU=">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</latexit> = dx

=

Z 2

0

0

@ 2(x + y) 32 3

3

0

1

A dy

<latexit sha1_base64="Bl93uybRlpUHU1BZ1sDdhJ4KiSk=">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</latexit>

=

Z 2

0

2(3 + y) 32 3

2(y) 32 3

!

dy

<latexit sha1_base64="vvUz1rC20Vg3q/JgTVO/87M/fAQ=">AAADSnicjVJNb9QwEJ2mhX7w0S0cuUSskLZCrLK7ldpLpapcOBbBtpWadpV4vVur2SRynEpRlB/Gf+APcOqV3hA31AvPUy8CCgJHsZ/fzBvPjB3niSpMEHxa8BaX7t1fXllde/Dw0eP11saTwyIrtZBDkSWZPo6jQiYqlUOjTCKPcy2jWZzIo/jitbUfXUpdqCx9b6pcns6iaaomSkQG1Kj1bjdUqRkFZ/0wkRPTCSc6EnW/M3hZbZ7Vt7tBU/ebpsH6am7+kzHUanpuNsfVqNUOugEP/y7oOdAmNw6y1hcKaUwZCSppRpJSMsAJRVTgO6EeBZSDO6UanAZSbJfU0Bq0JbwkPCKwF5in2J04NsXexixYLXBKgl9D6dMLaDL4aWB7ms/2kiNb9m+xa45pc6uwxi7WDKyhc7D/0s09/1dnazE0oR2uQaGmnBlbnXBRSu6Kzdz/qSqDCDk4i8ewa2DBynmffdYUXLvtbcT2z+xpWbsXzreka5elpEuOWv3IvuY7VLDn3MsKyGDmW8KT6P3+AO6Cw363N+gGb7fae/vucazQM3pOHbyAbdqjN3RAQ2Tzga6Q4bX30fvqffNubl29Bad5Sr+MxaXvQa673Q==</latexit>

= 4(3 + y) 52 15

4(y) 52 15

2

0

<latexit sha1_base64="F1szGqapc/QFC0hV2u77mM/HChI=">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</latexit>

= 4(5) 52 15

4(2) 52 15

<latexit sha1_base64="Kxt3KUp3iowrP5wmZLNrWQ54IXU=">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</latexit>

(17)

Exemple

u<latexit sha1_base64="XgJh78xj8/5HXM0hYmlnFbkmOvM=">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</latexit> = x dv<latexit sha1_base64="87hVmzQC26lkCttshxgJpectngA=">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</latexit> = exydx v = exy

y

<latexit sha1_base64="7bZyIfQjmP5HBjO0DkdaNn3ajz4=">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</latexit>

du<latexit sha1_base64="gNvm+ShnHnrF7be0i1nE2d5HyAU=">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</latexit> = dx Z 2

1

Z 1

0

xexy dxdy

<latexit sha1_base64="tYsqAYsu4zSKV9mx5QuD7u5Pjak=">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</latexit>

=

Z 2

1

✓ xexy y

Z 1

0

exy

y dx

dy

<latexit sha1_base64="v3vmh6nNpeZWcxiOZRzcpuze0Lc=">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</latexit>

=

Z 2

1

xexy y

exy y2

1 0

!

dy

<latexit sha1_base64="YQgDEnMrEZPXx0MjxFtRZ+jV/cQ=">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</latexit>

=

Z 2

1

✓ ey y

ey

y2 + 1 y2

dy

<latexit sha1_base64="yws0cRfl4Q38cgGCfy0fkyP9A1M=">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</latexit>

(18)

Exemple

u =

1 y

<latexit sha1_base64="fv+Jtm27Q69OHAyS8Lnpwz9h4sA=">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</latexit>

dv<latexit sha1_base64="8U5FdqkVi+0oftwZ4fQVZks7Pf8=">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</latexit> = eydy v<latexit sha1_base64="BJM/P76jx1khmzVkc4EA2G9vOUU=">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</latexit> = ey du = dy

y2

<latexit sha1_base64="csT/mG6BBj0B0/psG3QsUN1QtDE=">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</latexit>

Z ey

y dy = ey

y +

Z ey

y2 dy

<latexit sha1_base64="/Zds8PePrxKh5KEylqAwGfe28wg=">AAADMHicjVJda9RAFD1N/aj1a9VHX4KLIAhLtgrqg1D0xccK3bbQbUsyO1uHZpMwmRRC2N/if/A/+Fpf9Ul8U5/6EzxznYp1EZ2QyZlzz7m5d2ayKje1S5JPS9HyhYuXLq9cWb167fqNm71bt7fqsrFKj1SZl3YnS2udm0KPnHG53qmsTmdZrrezo5c+vn2sbW3KYtO1ld6bpYeFmRqVOlIHvWdjU7h4PLWp6vR+O+/a+aR9fm79cEGyv0bRQa+fDBIZ8SIYBtBHGBtl7xRjTFBCocEMGgUccY4UNZ9dDJGgIreHjpwlMhLXmGOV3oYqTUVK9ojzIVe7gS249jlrcSv+Jedr6Yxxn56SOkvs/xZLvJHMnv1b7k5y+tpafrOQa0bW4Q3Zf/nOlP/r8704TPFUejDsqRLGd6dClkZ2xVce/9aVY4aKnMcTxi2xEufZPsfiqaV3v7epxL+K0rN+rYK2wbdQpcaxZG1/Vd/JGRrGK9nLlshxllPilRj+eQEWwdbaYPhokLx+3F9/ES7HCu7iHh7wBjzBOl5hAyNW8xbvcYIP0bvoY/Q5+vJTGi0Fzx2cG9H3H4gxtD4=</latexit>

Z ey y

ey

y2 dy = ey y

<latexit sha1_base64="K612qsdYC+3FqdQarrmcyhJZKtQ=">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</latexit>

=

Z 2

1

✓ ey y

ey

y2 + 1 y2

dy

<latexit sha1_base64="yws0cRfl4Q38cgGCfy0fkyP9A1M=">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</latexit>

= ey y

1 y

2

1

<latexit sha1_base64="ZDu9aZbPJiAdLtvtsN1GO39AbLE=">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</latexit>

= e2 2

1

2 (e 1)

<latexit sha1_base64="uDcALQ82IjpFblHPpaIL94X1V9k=">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</latexit>

Z 2

1

Z 1

0

xexy dxdy

<latexit sha1_base64="tYsqAYsu4zSKV9mx5QuD7u5Pjak=">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</latexit>

(19)

Exemple

Z 2

1

Z 1

0

xexy dxdy

<latexit sha1_base64="tYsqAYsu4zSKV9mx5QuD7u5Pjak=">AAADFnicjVJBT9RAFP6oooigKx492LAx8bRpwQSORC4cMWGBhGU3bXeACd22aadkmw1H/wP/wateuRGvXvkHcOMn8M1zMAoxOk1nvvne+96892biItWVCYLLKe/R4+knT2eezT6fm3/xsvVqYbvK6zJR3SRP83I3jiqV6kx1jTap2i1KFY3iVO3Ex+vWvnOiykrn2ZZpCrU/ig4zfaCTyJAatN72dGYGYX9J1qAf+mNf9Sfj5rTnD8fDZtBqB51Ahv8QhA604cZm3rpBD0PkSFBjBIUMhjhFhIrfHkIEKMjtY0KuJNJiVzjFLLU1vRQ9IrLHnA+523Nsxr2NWYk64Skp/5JKH++oyelXEtvTfLHXEtmyf4s9kZg2t4Zr7GKNyBockf2X7s7zf3W2FoMDrEoNmjUVwtjqEhellq7YzP3fqjKMUJCzeEh7SZyI8q7Pvmgqqd32NhL7lXha1u4T51vj2mWpcCJRm1/ZT+QONe2F9LIhMpzllvgkwvsP4CHYXuqEy53g04f22kf3OGbwBot4zxewgjVsYBNdZvMZX/AV37wz79y78L7/dPWmnOY1/hjej1sMBKdI</latexit>

=

Z 1

0

Z 2

1

xexy dydx

<latexit sha1_base64="FG5DOfffw0Qagq1lpB6WZ6QN5Ak=">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</latexit>

=

Z 1

0

e2x ex dx

<latexit sha1_base64="1JyrAfWpAPO4yU5UBLglHLq9evo=">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</latexit>

(prise 2)

= e2x

2 ex

1

0

<latexit sha1_base64="vQkfDY2N46jISs9JyrCDG5Ld4HE=">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</latexit>

= e2 2

1

2 (e 1)

<latexit sha1_base64="uDcALQ82IjpFblHPpaIL94X1V9k=">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</latexit>

= e2

2 e ( 1

2 1)

<latexit sha1_base64="6SuTN0Utnncqdii6FC2DGg6I/2A=">AAADGnicjVLLThRBFD20DxBfoy7ddJyY4IJJN5rIxoTAxiUmDJAAmu6iBiv0dHeqq0kmnfkA/8F/YKtbdsatG/9AdnwCpy6FUYnR6lT1qXPvuXXvrcrrwjQuSb7PRNeu37g5O3dr/vadu/fu9x483Gyq1io9VFVR2e08a3RhSj10xhV6u7Y6G+eF3soP17x960jbxlTlhpvUem+cHZRmZFTmSL3r9V/FuyObqU6/7ZamU85FvbhwQaWyTZ/RKxkkMuKrIA2gjzDWq94ZdrGPCgotxtAo4YgLZGj47SBFgprcHjpylsiIXWOKeWpbeml6ZGQPuR5wtxPYknsfsxG14ikFp6UyxlNqKvpZYn9aLPZWInv2b7E7ielzm/Cfh1hjsg7vyf5Ld+n5vzpfi8MIy1KDYU21ML46FaK00hWfefxLVY4RanIe79NuiZUoL/sci6aR2n1vM7H/EE/P+r0Kvi1OQ5YaRxJ18jP7Tu7Q0F5LLydEjqvcEp9E+ucDuAo2lwbp80Hy5kV/ZTU8jjk8xhMs8AW8xApeYx1DZvMBx/iEz9HH6CT6En29cI1mguYRfhvRt3NTPqho</latexit>

La morale de cette histoire…

changer l’ordre d’intégration peut simplifier les calculs!

=

Z 1

0

exy 2

1

dx

<latexit sha1_base64="mkqvtApGTUBIY7n+VuAfhKsRqk4=">AAADIXicjVLLbtRAEKyYVwivBY5cLFZIgcPKTpDgghSFC8cgsUmkOLuyvbObUby2NR5HsZb8A//AP3CFKzfEDfEDcOMTqGkmCIgQjOVxTXVXu7uns7rQjY2iT0vBufMXLl5avrxy5eq16zd6N29tN1VrcjXMq6Iyu1naqEKXami1LdRubVQ6zwq1kx0+dfadI2UaXZUvbFer/Xk6K/VU56klNe49eJLo0o6jUZwUampXw1CNFsfdSbKpZy/H8WgtTIyeHdj7k+Nxrx8NIlnhWRB70IdfW1XvGxJMUCFHizkUSljiAikaPnuIEaEmt48FOUOkxa5wghVqW3opeqRkD7nPeNrzbMmzi9mIOudfCr6GyhD3qKnoZ4jd30KxtxLZsX+LvZCYLreO38zHmpO1OCD7L92p5//qXC0WUzyWGjRrqoVx1eU+SitdcZmHv1RlGaEm5/CEdkOci/K0z6FoGqnd9TYV+xfxdKw75963xVefpcKRRO1+Zr+QO9S019LLjshyl1viSMR/DsBZsL02iNcH0fOH/Y1NPxzLuIO7WOUEPMIGnmELQ2bzCm/wFu+C18H74EPw8YdrsOQ1t/HbCj5/B42iq1c=</latexit>

(20)

Faites les exercices suivants

p. 843 # 7 à 14

p. 843 # 7 à 12

(21)

Si f<latexit sha1_base64="Dka4dcZ0cpcHHw1s2s0dqUiCg24=">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</latexit> (x, y) = g(x)h(y)

Z d

c

Z b

a

f (x, y) dxdy =

Z b a

g(x) dx

! Z d c

h(y) dy

!

<latexit sha1_base64="oiEvHGlecgsmwBTN22k+HrTr3u0=">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</latexit>

alors

(22)

Exemple

=

✓Z 1

0

ex dx

◆ Z 2

0

cos y dy

!

<latexit sha1_base64="laeNCL9hO4fh1Btns6nsstYVdkU=">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</latexit>

Z 1

0

Z 2

0

ex cos y dydx

<latexit sha1_base64="YctkXoHANlTAmhqkNc4rA73YMKY=">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</latexit>

Z 1

0

Z 2

0

ex cos y dydx

<latexit sha1_base64="YctkXoHANlTAmhqkNc4rA73YMKY=">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</latexit>

=

Z 1

0

ex sin y

2

0 dx

<latexit sha1_base64="aN/kZd+Bl5BJvjRiFm5dNNGNBgY=">AAADI3icjVI9b9RAEH0xXyF8HVDSWJyQKNDJDkjQIEWhoQwSl0SKk5Pt2ztW8dnWeh3FMvcn+A/8B1po6RANBT10/ATeDhsERAjW8vrtm3njmdnJ6kI3Noo+rQRnzp47f2H14tqly1euXhtcv7HdVK3J1TivisrsZmmjCl2qsdW2ULu1UekiK9ROdvjE2XeOlGl0VT63Xa32F+m81DOdp5bUZHDvcaJLO4kO4lAdHCeNLsMu2dTzl6SSmUnzPqn1sl9fJuH0eDIYRqNIVngaxB4M4ddWNfiGBFNUyNFiAYUSlrhAiobPHmJEqMntoydniLTYFZZYo7all6JHSvaQ+5ynPc+WPLuYjahz/qXga6gMcYeain6G2P0tFHsrkR37t9i9xHS5dfxmPtaCrMULsv/SnXj+r87VYjHDI6lBs6ZaGFdd7qO00hWXefhLVZYRanIOT2k3xLkoT/ociqaR2l1vU7F/EU/HunPufVt89VkqHEnU7mf2vdyhpr2WXnZElrvcEkci/nMAToPt9VF8fxQ9ezDc2PTDsYpbuI27nICH2MBTbGHMbF7hDd7iXfA6eB98CD7+cA1WvOYmflvB5+9qGqy/</latexit>

=

Z 1

0

ex dx

<latexit sha1_base64="NVYJX4mQ6Z/lNsEvBvoAqKfiF8U=">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</latexit>

=<latexit sha1_base64="SbkvN6TJfbybMIOaz0q9rE/I7aY=">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</latexit> e 1

= (e<latexit sha1_base64="6XcsWUd9YE56iyvrWIAnOq/zJhA=">AAADGHicjVLBThRBEH2MqICiqxy9DGxM4MBmRkj0QkLkwhETFkhYQmaG3qXD7Mykp4dks+HMP/APXvXqzXj1xh/IzU/gddEYkRjtyfS8elWvpqq60yrXtY2iy4ngweTDR4+npmeePJ199rz14uVOXTYmU92szEuzlya1ynWhulbbXO1VRiXDNFe76cmG8++eKlPrsti2o0odDJNBofs6Syypw9b8Wi9XfbuoluOe0YNju3Rjx8uRtw9b7agTyQrvg9iDNvzaKls/0cMRSmRoMIRCAUucI0HNZx8xIlTkDjAmZ4i0+BXOMENtwyjFiITsCfcBrX3PFrRdzlrUGf+S8zVUhnhNTck4Q+z+Foq/kcyO/VvuseR0tY34TX2uIVmLY7L/0t1G/q/O9WLRxzvpQbOnShjXXeazNDIVV3n4W1eWGSpyDh/Rb4gzUd7OORRNLb272Sbi/yGRjnV25mMbXPkqFU4l6+hX9WM5Q01/JbMcEVnuckq8EvGfF+A+2HnTiVc60YfV9vp7fzmm8AoLWOQNeIt1bGILXVZzjo/4hM/BRfAl+Bp8uwkNJrxmDndW8P0airynaQ==</latexit> 1) (1 0) =<latexit sha1_base64="SbkvN6TJfbybMIOaz0q9rE/I7aY=">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</latexit> e 1

(23)

Faites les exercices suivants

p.843 # 29

(24)

Aujourd’hui, nous avons vu

Double somme de Riemann.

Intégrale partielle.

Intégrale itérée.

(25)

Devoir:

p. 843 # 1 à 22 et 25 à 31

p. 843 # 1 à 18 et 21 à 27

Références

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