3.2 INTÉGRALE DOUBLE SUR DOMAINES DIVERS

(1)

cours 18

3.2 INTÉGRALE DOUBLE SUR

DOMAINES DIVERS

(2)

Au dernier cours, nous avons vu

(3)

Au dernier cours, nous avons vu

Double somme de Riemann.

(4)

Au dernier cours, nous avons vu

Intégrale partielle.

(5)

Au dernier cours, nous avons vu

Intégrale partielle.

Intégrale itérée.

(6)

Aujourd’hui, nous allons voir

(7)

Aujourd’hui, nous allons voir

Intégrale double sur des régions non rectangulaires.

(8)

Aujourd’hui, nous allons voir

Intégrale double sur des régions non rectangulaires.

Région de Type I et II

(9)

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

R<latexit sha1_base64="7N2LeSiz0+W4o1VuQa1lUnFyEr8=">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</latexit>

(10)

(11)

Le truc est de se placer dans un rectangle

(12)

(13)

Le truc est de se placer dans un rectangle 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

(14)

Le truc est de se placer dans un rectangle F (x, y)

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

0 (x, y) 2/ R.

(15)

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.

(16)

a<latexit sha1_base64="HXNi2ziZI5xVOeyio5Y21rCNhno=">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</latexit> b<latexit sha1_base64="D+WRdc6TbX/xKWz3mmXpP6rTDt8=">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</latexit>

g(x)

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

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

Type I

(17)

a<latexit sha1_base64="HXNi2ziZI5xVOeyio5Y21rCNhno=">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</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

(18)

Exemple

f<latexit sha1_base64="mTcvjx3Ux6xWx8YO2AKY5WZotrQ=">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</latexit> (x, y) = 2x 3y

On veut intégrer la fonction

(19)

Exemple

Sur la région bornée par:

(20)

Exemple

y<latexit sha1_base64="v0h6vsUNOe8KdiBe6rkuYLrgnkY=">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</latexit> = x² + 1

(21)

Exemple

y<latexit sha1_base64="v0h6vsUNOe8KdiBe6rkuYLrgnkY=">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</latexit> = x² + 1 y<latexit sha1_base64="ALbAlonIV0EP94/lKOOCO/OdKnM=">AAAC+HicjVJNT9tAEH0Y2tL0gwDHXiyiSj1FTlupXCohuHAEQUikNKpss0lXcWzLXqO6ET+BK1y5oV77b/oPyo2f0LfTDSpFVbuW12/fzBvPzE6UJ7o0QfB9wVtcevDw0fLjxpOnz56vNFfXjsqsKmLVjbMkK/pRWKpEp6prtElUPy9UOI0S1YsmO9beO1FFqbP00NS5Gk7DcapHOg4NqYP6/eePzVbQDmT590HHgRbc2suaN/iAY2SIUWEKhRSGOEGIks8AHQTIyQ0xI1cQabErnKJBbUUvRY+Q7IT7mKeBY1OebcxS1DH/kvAtqPTxkpqMfgWx/Zsv9koiW/ZvsWcS0+ZW8xu5WFOyBp/I/ks39/xfna3FYIRNqUGzplwYW13solTSFZu5/1tVhhFychYf014Qx6Kc99kXTSm1296GYv8hnpa159j5Vrh2WSqcSNT6NvuZ3KGmPZde1kSGu9wSR6Lz5wDcB0ev25037WD/bWtr2w3HMl5gA684Ae+whV3soctsxjjDOS68L96ld+V9/eXqLTjNOu4s79tPBtacCA==</latexit> = x

(22)

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 f<latexit sha1_base64="mTcvjx3Ux6xWx8YO2AKY5WZotrQ=">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</latexit> (x, y) = 2x 3y

(23)

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

(24)

Exemple

(25)

Exemple

Z ₂

0

Z _x²₊₁

x

2x 3y dydx

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

(26)

Exemple

Z ₂

0

Z _x²₊₁

x

2x 3y dydx

=

Z ₂

0

2xy 3y² 2

x²+1 x

!

dx

<latexit sha1_base64="6T2kjerEPrOJYAY5+AjryqVN/E8=">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</latexit>

(27)

Exemple

Z ₂

0

Z _x²₊₁

x

2x 3y dydx

=

Z ₂

0

2xy 3y² 2

x²+1 x

!

dx

=

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>

(28)

Exemple

Z ₂

0

Z _x²₊₁

x

2x 3y dydx

=

Z ₂

0

2xy 3y² 2

x²+1 x

!

dx

=

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>

(29)

Exemple

Z ₂

0

Z _x²₊₁

x

2x 3y dydx

=

Z 2

0

2x³ + 2x 3(x⁴ + 2x² + 1) 2

x²

2 dx

(30)

Exemple

Z ₂

0

Z _x²₊₁

x

2x 3y dydx

=

Z 2

0

2x³ + 2x 3(x⁴ + 2x² + 1) 2

x²

2 dx

=

Z ₂

0

3x⁴

2 + 2x³ 7x²

2 + 2x 3

2 dx

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

(31)

Exemple

Z ₂

0

Z _x²₊₁

x

2x 3y dydx

=

Z ₂

0

3x⁴

2 + 2x³ 7x²

2 + 2x 3

2 dx

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