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3.2 INTÉGRALE DOUBLE SUR DOMAINES DIVERS

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Texte intégral

(1)

cours 18

3.2 INTÉGRALE DOUBLE SUR

DOMAINES DIVERS

(2)

Au dernier cours, nous avons vu

Double somme de Riemann.

Intégrale partielle.

Intégrale itérée.

(3)

Aujourd’hui, nous allons voir

Intégrale double sur des régions non rectangulaires.

Région de Type I et II

(4)

Comment faire pour intégrer une région qui n’est pas rectangulaire?

R<latexit sha1_base64="7N2LeSiz0+W4o1VuQa1lUnFyEr8=">AAAC93icjVJNT9tAEH1x+SqfaXvsxWqExCmy20rliODSI6AmQYII2WYDqzi2Za+RrIh/wJVee6t65efwD+itP6Fvp5uqLUJlLa/fvpk3npmduEh1ZYLgruU9m5tfWFx6vryyura+0X7xsl/ldZmoXpKneXkUR5VKdaZ6RptUHRWliiZxqgbxeM/aB5eqrHSefTJNoYaT6DzTI51EhtShf3ja7gTdQJb/EIQOdODWft7+gROcIUeCGhMoZDDEKSJUfI4RIkBBbogpuZJIi13hCsvU1vRS9IjIjrmf83Ts2IxnG7MSdcK/pHxLKn1sUpPTryS2f/PFXktkyz4WeyoxbW4Nv7GLNSFrcEH2f7qZ51N1thaDEbalBs2aCmFsdYmLUktXbOb+H1UZRijIWXxGe0mciHLWZ180ldRuexuJ/V48LWvPifOt8d1lqXApUZvf2U/lDjXthfSyITLc5ZY4EuG/A/AQ9N92w3fd4OB9Z2fXDccSXuMNtjgBH7CDj9hHj9mMcI0bfPYa74v31fv2y9VrOc0r/LW825/8bptC</latexit>

Le truc est de se placer dans un rectangle

ZZ

R

f (x, y)dA =

Z d

c

Z b

a

F (x, y) dxdy

<latexit sha1_base64="N6ccy5D64oiDyjvjSMnJtmrrVes=">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</latexit>

F (x, y)

(f (x, y) (x, y) 2 R

0 (x, y) 2/ R.

<latexit sha1_base64="heNAriBBi+pTTBRDL4AYDZDymqw=">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</latexit>

Et de considérer la fonction

(5)

a<latexit sha1_base64="HXNi2ziZI5xVOeyio5Y21rCNhno=">AAAC9nicjVJNS8NAEH2NX7V+VT16CRbBU0lV0GPRi8cW7AeoSJJudTFNQrIpluIv8KpXb+LVv+M/0Js/wdlxK2oR3ZDN2zfzJjOz48WBTJXjPOesicmp6Zn8bGFufmFxqbi80kyjLPFFw4+CKGl7bioCGYqGkioQ7TgRbs8LRMu7PND2Vl8kqYzCIzWIxWnPPQ9lV/quIqrunhVLTtnhZY+DigElmFWLim84QQcRfGToQSCEIhzARUrPMSpwEBN3iiFxCSHJdoFrFEibkZcgD5fYS9rP6XRs2JDOOmbKap/+EtCbkNLGBmki8ksI67/ZbM84smZ/iz3kmDq3AX09E6tHrMIFsX/pRp7/1elaFLrY4xok1RQzo6vzTZSMu6Izt79UpShCTJzGHbInhH1Wjvpssybl2nVvXba/sKdm9dk3vhleTZYCfY46+Mx+yHcoyR5zLweEFO18SzQSlZ8DMA6aW+XKdtmp75Sq+2Y48ljDOjZpAnZRxSFqaHA2N7jFnXVl3VsP1uOHq5UzmlV8W9bTO6qqmyc=</latexit> b<latexit sha1_base64="D+WRdc6TbX/xKWz3mmXpP6rTDt8=">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</latexit>

g(x)

<latexit sha1_base64="1ZLBwVgD5sCUD4yHL4cZSk9KwcU=">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</latexit>

Z b

a

Z h(x)

g(x)

f (x, y) dydx

<latexit sha1_base64="GozjyO7KQFWGOC54qUfiqgbaN3Y=">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</latexit>

h(x)<latexit sha1_base64="XhsvBssyl/h/UmV8C1FFbcMH/xQ=">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</latexit>

Type I

(6)

Exemple

y<latexit sha1_base64="v0h6vsUNOe8KdiBe6rkuYLrgnkY=">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</latexit> = x2 + 1 y<latexit sha1_base64="ALbAlonIV0EP94/lKOOCO/OdKnM=">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</latexit> = x x<latexit sha1_base64="Mph0riMHN7AtuCjQ2PoOV/+z/qE=">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</latexit> = 0 x<latexit sha1_base64="hI+6FdKbQfjBgn9LSgJhHW44Uoo=">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</latexit> = 2 f<latexit sha1_base64="mTcvjx3Ux6xWx8YO2AKY5WZotrQ=">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</latexit> (x, y) = 2x 3y

On veut intégrer la fonction

Sur la région bornée par:

Z 2

0

Z x2+1

x

2x 3y dydx

<latexit sha1_base64="QdwRHso94yddgmyad7U2cUfw8X8=">AAADGXicjVLBbtNAEH01BdIWaIAjF6sREhIispNKcKzgwrFIpK3UNJHtbMuqjm3Z6ypWlDv/wD9whSs3xJUTf0Bv/YS+nW4RUCG6lnffvpk3OzO7cZHqygTBjyXvxvLNW7dbK6trd+7eW2/ff7BT5XWZqEGSp3m5F0eVSnWmBkabVO0VpYqmcap24+NX1r57ospK59lb0xTqYBodZfpQJ5EhNW5vDHVmxsGoJ+tsNJ+Nek/DRW/2rN8M/UnjT2bjdifoBjL8qyB0oAM3tvP2GYaYIEeCGlMoZDDEKSJU/PYRIkBB7gBzciWRFrvCAqvU1vRS9IjIHnM+4m7fsRn3NmYl6oSnpPxLKn08pianX0lsT/PFXktky/4r9lxi2twarrGLNSVr8I7s/3SXntfV2VoMDvFCatCsqRDGVpe4KLV0xWbu/1aVYYSCnMUT2kviRJSXffZFU0nttreR2H+Kp2XtPnG+NU5dlgonErX5lf1c7lDTXkgvGyLDWW6JTyL8+wFcBTu9btjvBm82O1sv3eNo4RE28IQv4Dm28BrbGDCb9/iIT/jsffC+eF+9bxeu3pLTPMQfw/t+DmwYqBc=</latexit>

=

Z 2

0

2xy 3y2 2

x2+1 x

!

dx

<latexit sha1_base64="6T2kjerEPrOJYAY5+AjryqVN/E8=">AAADO3icjVJNb9NAEH01Hy3lowGOXCwipCJEZLuV4IJUUQ4ci0TaSnUT2ZtNuqpjW+t1FcvkD/Ef+A+9lisXWnHgwp3Z6bYCKgRref32zbzxzOykZaYqEwQnC9616zduLi7dWr595+69lc79B9tVUWsh+6LICr2bJpXMVC77RplM7pZaJtM0kzvp4aa17xxJXakif2+aUu5Pk0muxkokhqhh582rWOVmGAyiOJNjs8p7L5o1z+OxTkS71gyieRvNY60mB+bDcDZoZ4PoWeiIp7Hvj2bDTjfoBbz8qyB0oAu3torOd8QYoYBAjSkkchjCGRJU9OwhRICSuH20xGlCiu0ScyyTtiYvSR4JsYe0T+i059iczjZmxWpBf8no1aT08YQ0BflpwvZvPttrjmzZv8VuOabNraFv6mJNiTU4IPZfugvP/9XZWgzGeMk1KKqpZMZWJ1yUmrtiM/d/qcpQhJI4i0dk14QFKy/67LOm4tptbxO2f2VPy9qzcL41Tl2WEkcctbnMvuU7VGQvuZcNIUM73xKNRPjnAFwF21EvXOsF79a7G6/dcCzhER5jlSbgBTbwFlvoUzYfcYwTfPY+eV+8M+/buau34DQP8dvyfvwEAey2RA==</latexit>

=

Z 2

0

2x(x2 + 1) 3(x2 + 1)2

2 2x2 + 3x2

2 dx

<latexit sha1_base64="DWhoiRvzcmgQzeFcmt/gDWkBV9o=">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</latexit>

=

Z 2

0

2x3 + 2x 3(x4 + 2x2 + 1) 2

x2

2 dx

<latexit sha1_base64="QWM4tD9H7pWGAqSwifOKP6Z2sXk=">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</latexit>

(7)

Exemple

y<latexit sha1_base64="v0h6vsUNOe8KdiBe6rkuYLrgnkY=">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</latexit> = x2 + 1 y<latexit sha1_base64="ALbAlonIV0EP94/lKOOCO/OdKnM=">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</latexit> = x x<latexit sha1_base64="Mph0riMHN7AtuCjQ2PoOV/+z/qE=">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</latexit> = 0 x<latexit sha1_base64="hI+6FdKbQfjBgn9LSgJhHW44Uoo=">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</latexit> = 2 f<latexit sha1_base64="mTcvjx3Ux6xWx8YO2AKY5WZotrQ=">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</latexit> (x, y) = 2x 3y

On veut intégrer la fonction

Sur la région bornée par:

Z 2

0

Z x2+1

x

2x 3y dydx

<latexit sha1_base64="QdwRHso94yddgmyad7U2cUfw8X8=">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</latexit>

=

Z 2

0

2x3 + 2x 3(x4 + 2x2 + 1) 2

x2

2 dx

<latexit sha1_base64="QWM4tD9H7pWGAqSwifOKP6Z2sXk=">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</latexit>

=

Z 2

0

3x4

2 + 2x3 7x2

2 + 2x 3

2 dx

<latexit sha1_base64="uSpQrTCmPMjN5ImLJGKaDVdnq2c=">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</latexit>

(8)

Exemple

y<latexit sha1_base64="v0h6vsUNOe8KdiBe6rkuYLrgnkY=">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</latexit> = x2 + 1 y<latexit sha1_base64="ALbAlonIV0EP94/lKOOCO/OdKnM=">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</latexit> = x x<latexit sha1_base64="Mph0riMHN7AtuCjQ2PoOV/+z/qE=">AAAC+HicjVLBTttAEH0YSiktJaXHXiyiSj1FDq1ULpWicuFIBYFIASHbbMIKx7bsdVQ36if0Ctfeql77N/wB3PiEvp1uUAEhWMvrt2/mjWdmJ8oTXZogOJ/xZueezD9deLb4/MXSy+XGq5XdMquKWHXjLMmKXhSWKtGp6hptEtXLCxWOokTtRScb1r43VkWps3TH1Lk6GIXDVA90HBpS218/BYeNZtAKZPl3QduBJtzayhpX2McRMsSoMIJCCkOcIETJp482AuTkDjAhVxBpsSt8xyK1Fb0UPUKyJ9yHPPUdm/JsY5aijvmXhG9BpY+31GT0K4jt33yxVxLZsvfFnkhMm1vNb+RijcgaHJN9SDf1fKzO1mIwwLrUoFlTLoytLnZRKumKzdz/ryrDCDk5i49oL4hjUU777IumlNptb0OxX4inZe05dr4VLl2WCmOJWl9nP5E71LTn0suayHCXW+JItG8PwF2wu9Zqv28FXz40O5/dcCzgDVbxjhPwER1sYgtdZjPED5zizPvm/fR+eb//uXozTvMaN5b35y9LI5u/</latexit> = 0 x<latexit sha1_base64="hI+6FdKbQfjBgn9LSgJhHW44Uoo=">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</latexit> = 2 f<latexit sha1_base64="mTcvjx3Ux6xWx8YO2AKY5WZotrQ=">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</latexit> (x, y) = 2x 3y

On veut intégrer la fonction

Sur la région bornée par:

Z 2

0

Z x2+1

x

2x 3y dydx

<latexit sha1_base64="QdwRHso94yddgmyad7U2cUfw8X8=">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</latexit>

=

Z 2

0

3x4

2 + 2x3 7x2

2 + 2x 3

2 dx

<latexit sha1_base64="uSpQrTCmPMjN5ImLJGKaDVdnq2c=">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</latexit>

= 3x5

10 + 2x4 4

7x3

6 + x2 3x 2

2

0

<latexit sha1_base64="HNZv6dnKWDkKjg1FYFBn8GKo+kY=">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</latexit>

= 3(16)

5 + 8 7(4)

3 + 4 3

<latexit sha1_base64="m0ZgBFZ3N4UsdgaKxUjCe3gjiI0=">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</latexit>

= 9(16)

15 + 9(15) 15

7(20) 15

<latexit sha1_base64="PCRUcrkGcIJoOsMxw0WNZXHGP6Y=">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</latexit>

= 149 15

<latexit sha1_base64="pWE9UJENCFruk7HeACRSqgho+EM=">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</latexit>

(9)

g(y)

<latexit sha1_base64="oJuci9pdbjXRlpj0vdcSLl0fdC8=">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</latexit>

c<latexit sha1_base64="kIj3PCo/PnukiEh+BqzQbgGcego=">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</latexit>

d<latexit sha1_base64="AgEZUY3o/taocyHM1Ly/jPafatk=">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</latexit>

Z d

c

Z h(y)

g(y)

f (x, y) dxdy

<latexit sha1_base64="fCs33CHxjWv2aAWk5c5L6sF/LSI=">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</latexit>

h(y<latexit sha1_base64="MHpRSOQvSH6MwEz7NxPUv4U/huY=">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</latexit> ) Type II

(10)

Exemple

f (x, y) = xy

<latexit sha1_base64="RJekzzdxyFxmI6PhoNQvNC5mcFc=">AAAC/nicjVJNSxxBEH2OiRqjZo3HXIYsgkJYZjWgF0HixaMBVw0qMjP2auN8MdMjDouQ/5CrXr1Jrvkr/gNz8yfkdaUNSUS0h+l5/apeTVV1RUWiKxMEN0Pe8IuXI6Njr8ZfT0xOvWlNv92q8rqMVS/Ok7zcicJKJTpTPaNNonaKUoVplKjt6GTN2rdPVVnpPNs0TaH20/Ao030dh4bUl/7c2YdmfuWsOWi1g04gy38Iug604dZG3rrDHg6RI0aNFAoZDHGCEBWfXXQRoCC3jwG5kkiLXeEc49TW9FL0CMmecD/iadexGc82ZiXqmH9J+JZU+pilJqdfSWz/5ou9lsiWfSz2QGLa3Bp+IxcrJWtwTPYp3b3nc3W2FoM+lqUGzZoKYWx1sYtSS1ds5v5fVRlGKMhZfEh7SRyL8r7Pvmgqqd32NhT7rXha1p5j51vjp8tS4VSiNn+yH8gdatoL6WVDZLjLLXEkuv8PwEOwtdDpLnaCzx/bq5/ccIzhHd5jjhOwhFWsYwM9ZpPiGy5w6X31rrxr7/tvV2/IaWbwz/J+/AKoAZ4Y</latexit>

x = y2

<latexit sha1_base64="nIevXO1+K1nRFzUVlAJQNjMcrzs=">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</latexit> x = 2y<latexit sha1_base64="NEPjWGYnP6moGjGPEgs1IokeLY4=">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</latexit>

y<latexit sha1_base64="BQ5eMn5KIdy3ToaaiewUTmqU55Y=">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</latexit> 2 = 2y

0 = y2 2y

<latexit sha1_base64="SP7tAHepwz7bxxc/e5Ah8GKJn4k=">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</latexit> = y(y 2)<latexit sha1_base64="AvEG4rNsXHaE8spB7UL07hxqrnc=">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</latexit>

= 8 16 3

<latexit sha1_base64="s09a5Ff4Ixr42L8CZDH/WyKdFG4=">AAADB3icjVLLbtRAEKyYVxJeJhy5WKyQuLCySQS5IEXhwjFIbBIpiSJ7MhtG8drWeLzSytoP4B+4wpUb4spn5A/CjU9ITTNBQIRgLI9rqrva3T1dNKVpXZqeLkRXrl67fmNxafnmrdt37sb3VrbburNKj1Rd1na3yFtdmkqPnHGl3m2szidFqXeKk5fevjPVtjV19cbNGn0wyY8rMzYqd6QO4/hFsv5kf2xz1WfP5v3q/DAepMNUVnIZZAEMENZWHX/HPo5QQ6HDBBoVHHGJHC2fPWRI0ZA7QE/OEhmxa8yxTG1HL02PnOwJ92Oe9gJb8exjtqJW/EvJ11KZ4BE1Nf0ssf9bIvZOInv2b7F7ielzm/FbhFgTsg5vyf5Ld+H5vzpfi8MY61KDYU2NML46FaJ00hWfefJLVY4RGnIeH9FuiZUoL/qciKaV2n1vc7Gfiadn/VkF3w7fQpYaU4k6+5l9L3doaG+klzMix11uiSOR/TkAl8H202G2Okxfrw02NsNwLOIBHuIxJ+A5NvAKWxgxmyne4wM+Ru+iT9Hn6MsP12ghaO7jtxV9PQcw26DR</latexit>

= 8 3

<latexit sha1_base64="hTf5MwV/9vzZBi50yWbM1wDK+Go=">AAADAnicjVJNTxRBEH2MH6yAuOjRy8QNCafNrJDIxYTgxeOauEACGzLT9GKH2Zmxp4dkM9kb/8GrXr0Zr/4R/4He/Am+LhujEoI9mZ7Xr+rVVFVXVuWmdknydSG6dfvO3cXOvaXllfurD7prD/fqsrFKj1SZl/YgS2udm0KPnHG5PqisTqdZrvezsxfevn+ubW3K4rWbVXo8TU8LMzEqdaTGz+OjiU1Vuz1vN+fH3V7ST2TFV8EggB7CGpbdHzjCCUooNJhCo4AjzpGi5nOIARJU5MZoyVkiI3aNOZaobeil6ZGSPeN+ytNhYAuefcxa1Ip/yflaKmOsU1PSzxL7v8VibySyZ6+L3UpMn9uM3yzEmpJ1eEP2Jt2l5//qfC0OE2xLDYY1VcL46lSI0khXfObxH1U5RqjIeXxCuyVWorzscyyaWmr3vU3F/k08PevPKvg2+B6y1DiXqLPf2bdyh4b2Sno5I3Lc5ZY4EoN/B+Aq2HvaH2z2k1dbvZ3dMBwdPMYTbHACnmEHLzHEiNm8xTu8x4foIvoYfYo+/3KNFoLmEf5a0ZefoDGf7g==</latexit>

= 24 3

16 3

<latexit sha1_base64="FtRov1GQkvw/V1fL7OAEbhDUor4=">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</latexit>

Z 2

0

Z 2y

y2

xy dxdy

<latexit sha1_base64="bcAhV7+WnWiNMUPVYPYV5jsSkKc=">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</latexit>

=

Z 2

0

x2y 2

2y y2

!

dy

<latexit sha1_base64="UG9883oQgUmfd6VIfeFbNyjxemQ=">AAADOHicjVJBT9RAFP6oCgiIix65NGxI4LLpVhO5kKBePELiAgllN+3s7DKh2zbTKaEp+3v8D/4Hrxpv3pSb8Rfw5jEYlRiYptNvvve+1/fevKRIVWmC4OuM9+Dho9m5+ccLi0tPlp+2Vp7tl3mlheyJPM31YRKXMlWZ7BllUnlYaBlPklQeJKdvrf3gTOpS5dl7UxfyeBKPMzVSIjZEDVqvtyOVmUHQD6NUjswG751opGPRnPfDetqE00ir8Ym5GDR1P5z2GyKvmc3I94f1oNUOOgEv/zboOtCGW7t56yciDJFDoMIEEhkM4RQxSnqO0EWAgrhjNMRpQortElMskLYiL0keMbGntI/pdOTYjM42ZslqQX9J6dWk9LFOmpz8NGH7N5/tFUe27P9iNxzT5lbTN3GxJsQanBB7l+7G8746W4vBCFtcg6KaCmZsdcJFqbgrNnP/j6oMRSiIs3hIdk1YsPKmzz5rSq7d9jZm+w/2tKw9C+db4dJlKXHGUevf2Td8h4rsBfeyJmRo51uikej+OwC3wX7Y6b7oBHsv2ztv3HDMYxVr2KAJeIUdvMMuepTNB3zCZ3zxPnrfvO/e5bWrN+M0z/HX8n5dAchOtew=</latexit>

=

Z 2

0

2y3 y5

2 dy

<latexit sha1_base64="7sfIidVqd9b2u4VFrXqqg/1/4CE=">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</latexit>

= y4 2

y6 12

2

0

<latexit sha1_base64="HzxN1tVO+tWDS8098o8TyPmsL0s=">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</latexit>

On veut intégrer la fonction Sur la région bornée par:

(11)

Exemple

y = p x

<latexit sha1_base64="Al+sfnnWRE5us6Z3WeACwuiqhSk=">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</latexit>

y = x 2

<latexit sha1_base64="VScS0hnXk+OVh0Ptuo2bfpDfhEk=">AAADAnicjVJNTxRBEH0MfiB+rXD0MnFj4mkziyR6MSF64YiJCySwMTNNL3aYnRl7eoiTyd78D17hys149Y/wD+TGT+B10RiVGO3J9Lx+Va+mqrqyKje1S5LTuWj+xs1btxfuLN69d//Bw96jpc26bKzSI1Xmpd3O0lrnptAjZ1yutyur02mW663s4I23bx1qW5uyeOfaSo+n6X5hJkaljtS4fbU7sanqPs26ldn7Xj8ZJLLi62AYQB9hbZS9c+xiDyUUGkyhUcAR50hR89nBEAkqcmN05CyREbvGDIvUNvTS9EjJHnDf52knsAXPPmYtasW/5HwtlTGeUlPSzxL7v8VibySyZ/8Wu5OYPreW3yzEmpJ1+ED2X7orz//V+VocJngpNRjWVAnjq1MhSiNd8ZnHv1TlGKEi5/Ee7ZZYifKqz7Foaqnd9zYV+w/x9Kw/q+Db4CxkqXEoUduf2Xdyh4b2SnrZEjnuckscieGfA3AdbK4Mhs8HydvV/trrMBwLeIwneMYJeIE1rGMDI2bzEV9whOPoc3QSfY2+XbpGc0GzjN9W9P0CLA2ghg==</latexit>

Z 4

0

Z px

x 2

xy dydx

<latexit sha1_base64="6h1JtiQcTLezNpLh5G9XdLeui14=">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</latexit>

=

Z 4

0

xy2 2

px

x 2

!

dx

<latexit sha1_base64="DZHgsdQyMJays97LDgSKl7ugwpo=">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</latexit>

=

Z 4

0

x2 2

x3

8 dx

<latexit sha1_base64="/C3qVTtQ1W+9j0qkYcMn3Kr30VU=">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</latexit>

= x3 6

x4 32

4

0

<latexit sha1_base64="I6YbZhCE6tRUbqInP1wprltfA/k=">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</latexit>

= 32

3 8

<latexit sha1_base64="B0muw1b+3PB6CyGjUxqQhGtBN7U=">AAADBnicjVJNT9tAEH2Yz4ZCAj32YhFV4tLISZDggoTaC8cgNQEpQcheNrCKY1vrNVJk5c5/6LW99lb1yt/oP4AbP6Gz003FhxBdy+u3b+aNZ2YnymKVmyD4PefNLywuLa+8qay+XVuv1jY2e3laaCG7Io1TfRKFuYxVIrtGmVieZFqG4yiWx9Hos7UfX0mdqzT5YiaZPB2HF4kaKhEaos5q1f3BUIeibLemZXv6ce+sVg8aAS//OWg6UIdbnbR2jwHOkUKgwBgSCQzhGCFyevpoIkBG3ClK4jQhxXaJKSqkLchLkkdI7Ij2Czr1HZvQ2cbMWS3oLzG9mpQ+PpAmJT9N2P7NZ3vBkS37UuySY9rcJvSNXKwxsQaXxL6mm3n+r87WYjDEHtegqKaMGVudcFEK7orN3H9QlaEIGXEWn5NdExasnPXZZ03Otdvehmy/ZU/L2rNwvgXuXJYSVxx18i/7ku9QkT3jXk4IGdr5lmgkmk8H4DnotRrNdiM42qkffHLDsYL32MI2TcAuDnCIDro8JV/xDd+9a++H99P79dfVm3Oad3i0vJs/tXGgpQ==</latexit>

= 32 3

24 3

<latexit sha1_base64="gRw7up458s6uTDJAEHUn9vYG/58=">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</latexit>

= 8 3

<latexit sha1_base64="P3eKC7s6/5iKEtNs7oxzQnhXIB8=">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</latexit>

f<latexit sha1_base64="RJekzzdxyFxmI6PhoNQvNC5mcFc=">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</latexit> (x, y) = xy

x = y2

<latexit sha1_base64="nIevXO1+K1nRFzUVlAJQNjMcrzs=">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</latexit> x = 2y<latexit sha1_base64="NEPjWGYnP6moGjGPEgs1IokeLY4=">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</latexit>

On veut intégrer la fonction Sur la région bornée par:

(12)

Faites les exercices suivants

p.850 # 1 à 10 p.851 # 1 à 10

(13)

R<latexit sha1_base64="sMDugpUFqPaDx4E9mwuJhOctWoQ=">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</latexit>

I<latexit sha1_base64="+CJFiQiiu0ypeQbNffDI6pNFoTQ=">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</latexit>

II<latexit sha1_base64="bV+W36N0rQDjQPeWvghjblBbTL8=">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</latexit>

II

<latexit sha1_base64="bV+W36N0rQDjQPeWvghjblBbTL8=">AAAC93icjVJNb9NAEH0xpYR+Bjj2YjWq1FPkQKVyrOiF3Apq2kqhQrazCas4tmWvK1kR/6BXuHJDXPk5/QftrT+Bt8Om6ocqWMvrt2/mjWdmJ8oTXZoguGh4TxaeLj5rPl9aXlldW2+9eHlUZlURq36cJVlxEoWlSnSq+kabRJ3khQqnUaKOo8m+tR+fqaLUWXpo6lydTsNxqkc6Dg2pj73e51Y76ASy/Ieg60Abbh1krWt8whAZYlSYQiGFIU4QouQzQBcBcnKnmJEriLTYFb5iidqKXooeIdkJ9zFPA8emPNuYpahj/iXhW1DpY4uajH4Fsf2bL/ZKIlv2sdgziWlzq/mNXKwpWYMvZP+lm3v+r87WYjDCW6lBs6ZcGFtd7KJU0hWbuX+rKsMIOTmLh7QXxLEo5332RVNK7ba3odgvxdOy9hw73wpXLkuFM4la32Q/kzvUtOfSy5rIcJdb4kh07w/AQ3D0utN90wk+7LT33rnhaGIDm9jmBOxiD+9xgD6zGeEc3/Ddq70f3k/v119Xr+E0r3Bneb//AE7mm2I=</latexit>

I<latexit sha1_base64="+CJFiQiiu0ypeQbNffDI6pNFoTQ=">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</latexit>

Parfois, une région n’est ni type I ni type II.

Mais on peut la découper en régions qui le sont.

(14)

Exemple

(1,<latexit sha1_base64="7rnErj3nL8CJcLuwIg6LHUTYj0w=">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</latexit> 0)

(0, 1)

<latexit sha1_base64="z+ZpPQi6tfx5e7svilX2PDg2Cqs=">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</latexit> (2, 1)<latexit sha1_base64="2SHiISk7zfGT/P1dHPh2Xs2ix70=">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</latexit> (1, 2)<latexit sha1_base64="Qr5mlQfbDPdu98J4Aky6p3yOQHQ=">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</latexit>

y = x + 1

<latexit sha1_base64="OmTm0I1mmiEFrD255CzzXx1Uq6Y=">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</latexit>

x<latexit sha1_base64="Mph0riMHN7AtuCjQ2PoOV/+z/qE=">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</latexit> = 0 x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1

y = x 1

<latexit sha1_base64="A8lt0irqZjdDhPdOBlFxlnTN9JE=">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</latexit>

Type I

Type II y<latexit sha1_base64="5TbN7xVtvPYqwd+NPCCMGwgHskA=">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</latexit> = x + 1

y<latexit sha1_base64="+pUELiRsRAaCjxdcXWgllLWbwv8=">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</latexit> = x + 3 x<latexit sha1_base64="hI+6FdKbQfjBgn9LSgJhHW44Uoo=">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</latexit> = 2 x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1

x<latexit sha1_base64="XxvdrhhqgYACdI1gFS44d5zkjNc=">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</latexit> = y + 1 x<latexit sha1_base64="yw8ssXnlc/1rbrHpyyj0l5NPZRk=">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</latexit> = y + 1

y<latexit sha1_base64="3Z2rGZii132jXVyMb9BKpxgH0qg=">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</latexit> = 0 y = 1

<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">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</latexit>

x = y 1

<latexit sha1_base64="u6lSuHT+z6forvhXUHBZlWQNVQE=">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</latexit>

x<latexit sha1_base64="lWaQf+kM+qULCsMQ6OGvBmIql90=">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</latexit> = y + 3

y<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">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</latexit> = 1 y<latexit sha1_base64="kQcg/ch0xdVy1wewEHAYoOcuTpU=">AAAC+HicjVLLTtxAECwMhGeSDRy5WKyQOK28BIlckFBy4QiCBSRAkW2GzQivbdnjlcyKT8g1uXJDXPkb/gBufAI1zYB4CCVjeVxT3dXu7ukoT3RpguB6yBseGf0wNj4xOTX98dPnxpeZnTKrilh14izJir0oLFWiU9Ux2iRqLy9U2IsStRud/LD23b4qSp2l26bO1WEv7Kb6WMehIbVVry79bDSDViDLfwvaDjTh1kbWuMMBjpAhRoUeFFIY4gQhSj77aCNATu4QA3IFkRa7whkmqa3opegRkj3h3uVp37EpzzZmKeqYf0n4FlT6WKAmo19BbP/mi72SyJZ9L/ZAYtrcan4jF6tH1uAX2X/pHj3/V2drMTjGN6lBs6ZcGFtd7KJU0hWbuf+sKsMIOTmLj2gviGNRPvbZF00ptdvehmK/EU/L2nPsfCvcuiwV+hK1fsp+IHeoac+llzWR4S63xJFovx6At2BnqdX+2go2l5tr391wjGMO81jkBKxgDevYQIfZdPEbf/DXO/XOvQvv8sHVG3KaWbxY3tU9Utubwg==</latexit> = 2 f (x, y) = x

y + 1

<latexit sha1_base64="GjYMhqiwS28nGn4bz1B6/E4Exn0=">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</latexit>

sur le carré de sommets

(15)

Exemple

Type I

y = x + 1

<latexit sha1_base64="OmTm0I1mmiEFrD255CzzXx1Uq6Y=">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</latexit>

x<latexit sha1_base64="Mph0riMHN7AtuCjQ2PoOV/+z/qE=">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</latexit> = 0 x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1

y = x 1

<latexit sha1_base64="A8lt0irqZjdDhPdOBlFxlnTN9JE=">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</latexit>

y<latexit sha1_base64="5TbN7xVtvPYqwd+NPCCMGwgHskA=">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</latexit> = x + 1

y<latexit sha1_base64="+pUELiRsRAaCjxdcXWgllLWbwv8=">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</latexit> = x + 3 x<latexit sha1_base64="hI+6FdKbQfjBgn9LSgJhHW44Uoo=">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</latexit> = 2 x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1 f (x, y) = x

y + 1

<latexit sha1_base64="GjYMhqiwS28nGn4bz1B6/E4Exn0=">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</latexit>

Z 1

0

Z x+1

x+1

x

y + 1 dydx +

Z 2

1

Z x+3

x 1

x

y + 1 dydx

<latexit sha1_base64="Sx2g4h/SKUrsP5l5zwpPPkfARmk=">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</latexit>

=

Z 1

0

x ln |y + 1| x+1

x+1

dx +

Z 2

1

x ln |y + 1| x+3

x 1

dx

<latexit sha1_base64="LREPvE267NWim+OwnMbYhM6ynLY=">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</latexit>

=

Z 1

0

(x ln | x + 2| x ln |x + 2|) dx +

Z 2

1

(x ln |y| x ln | x + 4|) dx

<latexit sha1_base64="fyUugnxsUJhkATESKXq9Ee1WH7k=">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</latexit>

ouin… essayons en changent l’ordre d’intégration

Références

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