3.1 INTÉGRALE DOUBLE

(1)

cours 17

3.1 INTÉGRALE

DOUBLE

(2)

Aujourd’hui, nous allons voir

(3)

Aujourd’hui, nous allons voir

Double somme de Riemann.

(4)

Aujourd’hui, nous allons voir

Intégrale partielle.

(5)

Aujourd’hui, nous allons voir

Intégrale partielle.

Intégrale itérée.

(6)

Sommes de Riemann

(7)

Sommes de Riemann

(8)

nlim!1

Xn

i=1

f (x^⇤_i ) x_i

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Sommes de Riemann

(9)

nlim!1

Xn

i=1

f (x^⇤_i ) x_i

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=

Z b

a

f (x) dx

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Sommes de Riemann

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

nlim!1 m!1

Xn

i=1

Xm

k=1

f (x^⇤_i , y_k^⇤) A_ik

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(19)

nlim!1 m!1

Xn

i=1

Xm

k=1

f (x^⇤_i , y_k^⇤) A_ik

=

ZZ

R

(x, y)dA

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(20)

nlim!1 m!1

Xn

i=1

Xm

k=1

f (x^⇤_i , y_k^⇤) A_ik

=

ZZ

R

(x, y)dA

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L’intégrale double

(21)

On peut déduire certaines propriétés de l’intégrale double définie

(22)

1)

(23)

Aire(R) = 0

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Si 1)

(24)

Aire(R) = 0

Si

ZZ

R

f (x, y)dA = 0

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alors 1)

(25)

Aire(R) = 0

Si

ZZ

R

f (x, y)dA = 0

alors 1)

ZZ

R

1dA = Aire(R)

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2)

(26)

Aire(R) = 0

Si

ZZ

R

f (x, y)dA = 0

alors 1)

ZZ

R

1dA = Aire(R)

2)

3) Linéarité

(27)

Aire(R) = 0

Si

ZZ

R

f (x, y)dA = 0

alors 1)

ZZ

R

1dA = Aire(R)

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

(28)

3) Décomposition de la région

(29)

R<latexit sha1_base64="qSgwlcn2bjS52UBt57Iu1ni6ZTQ=">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</latexit> = R₁ [ R₂ [ · · · [ R_n 3) Décomposition de la région

(30)

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

=

[n

k=1

R_k

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(31)

=

[n

k=1

R_k

8i 6= j

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(32)

=

[n

k=1

R_k

8i 6= j

<latexit sha1_base64="T1X+9XNAnZJVhka4z+pMsZA/L0k=">AAADBnicjVLBThRBEH2MCAsqrHLUkAmExNNmFg963OjFIyQukLCEzAy92NI7M/b0kGw2HEn4B69y5UbgyG/4BxIufIKvi8GohEhPpuf1q3o1VdWVFEaXLop+jAWPxh9PTDampp88fTYz23z+Yq3MK5uqbpqb3G4kcamMzlTXaWfURmFVPEiMWk/2Pnj7+r6ypc6zT25YqK1BvJvpvk5jR2q7Odvr5zY2JtS9TH0Nv2w3F6NWJCu8C9o1WOy8Ol+9Opw/X8mb1+hhBzlSVBhAIYMjNohR8tlEGxEKclsYkbNEWuwKB5imtqKXokdMdo/7Lk+bNZvx7GOWok75F8PXUhliiZqcfpbY/y0UeyWRPXtf7JHE9LkN+U3qWAOyDp/J/k936/lQna/FoY93UoNmTYUwvrq0jlJJV3zm4R9VOUYoyHm8Q7slTkV52+dQNKXU7nsbi/2neHrWn9Pat8JlnaXCvkQd/s5+JHeoaS+kl0Mix11uiSPR/ncA7oK15Vb7TSta5Wy8x81q4CUW8JoT8BYdfMQKujIl3/Adx8FRcBKcBmc3rsFYrZnDXyu4+AWcFaS5</latexit> R_i \ R_j = ;<latexit sha1_base64="I4St4XeBUh8ta5ajgWPekn759Oo=">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</latexit>

(33)

=

[n

k=1

R_k

8i 6= j

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

ZZ

R

f (x, y)dA =

Xn

k=1

ZZ

R_k

f (x, y)dA

<latexit sha1_base64="oiKVzgqPzlYIx3rRdFR9bybyfCA=">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</latexit>

(34)

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

(35)

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

(36)

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?

(37)

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.

(38)

Exemple

(39)

Exemple

Z

xy + y²dx

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

(40)

Exemple

Z

xy + y²dx

= xy²

2 + y²x + C(y)

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

(41)

Une fonction qui dépend seulement de y<latexit sha1_base64="ADnGsQtcomAbrlTQt0snNnnV3IQ=">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</latexit>

Exemple

Z

xy + y²dx

= xy²

2 + y²x + C(y)

(42)

Exemple

Z

xy + y²dx

= xy²

2 + y²x + C(y)

Z

xy + y²dy

(43)

Exemple

Z

xy + y²dx

= xy²

2 + y²x + C(y)

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>

(44)

Une fonction qui dépend seulement de x<latexit sha1_base64="HA6BPTQP3UBB+d9iHJHQ4r2LiZk=">AAAC9nicjVJNS8NAEH3G7++qRy/BIngqiQp6FL14bMFqQUWSdFuXpklINqWh+Au86tWbePXv+A/05k9wdtyKH4huyObtm3mTmdnxk1BmynGeRqzRsfGJyanpmdm5+YXF0tLycRbnaSDqQRzGacP3MhHKSNSVVKFoJKnwun4oTvzOgbaf9ESayTg6UkUizrteO5ItGXiKqFr/olR2Kg4v+ydwDSjDrGpcesUZmogRIEcXAhEU4RAeMnpO4cJBQtw5BsSlhCTbBa4wQ9qcvAR5eMR2aG/T6dSwEZ11zIzVAf0lpDclpY110sTklxLWf7PZnnNkzf4We8AxdW4FfX0Tq0uswiWxf+mGnv/V6VoUWtjlGiTVlDCjqwtMlJy7ojO3P1WlKEJCnMZNsqeEA1YO+2yzJuPadW89tj+zp2b1OTC+OV5MlgI9jlp8ZD/gO5RkT7iXBSFFO98SjYT7fQB+guPNirtVcWrb5b19MxxTWMUaNmgCdrCHQ1RR52yucYNbq2/dWffWw7urNWI0K/iyrMc35cibPg==</latexit>

Exemple

Z

xy + y²dx

= xy²

2 + y²x + C(y)

Z

xy + y²dy

<latexit sha1_base64="F2ZPWV3L5GLg40pyjXcBXPSbxIA=">AAADAnicjVLLTtxAECwcCM+QTXLkYrFCQkJaeQEpHFG45AgSu4sEC7K9A4zw2sYeo1ir3PiHXJNrbhHX/Ah/kNzyCalpBsRDERnL45rqrnZ3T0d5oksTBNdj3ovxiZeTU9Mzs3Ov5l833rztlllVxKoTZ0lW7EVhqRKdqo7RJlF7eaHCYZSoXnS2Ze29C1WUOkt3TZ2r/jA8SfWxjkNDqn+gU+N/qlfqw9VBfdRoBq1Alv8UtB1owq3trPEHBxggQ4wKQyikMMQJQpR89tFGgJxcHyNyBZEWu8JnzFBb0UvRIyR7xv2Ep33HpjzbmKWoY/4l4VtQ6WOJmox+BbH9my/2SiJb9l+xRxLT5lbzG7lYQ7IGp2Sf0916/q/O1mJwjA2pQbOmXBhbXeyiVNIVm7l/ryrDCDk5iwe0F8SxKG/77IumlNptb0Ox/xJPy9pz7Hwr/HZZKlxI1Pou+5HcoaY9l17WRIa73BJHov14AJ6C7mqrvdYKdtabmx/ccExhAYtY5gS8xyY+YhsdZnOOL/iKb96l99374V3duHpjTvMOD5b38y/jh6AH</latexit>

= xy²

2 + y³

3 + C(x)

<latexit sha1_base64="gYgzaSQuqlH8N0eO+vVQHdl+OUw=">AAADGnicjVLLbtRAEKyYR0J4ZIEjF4sVUlCklXcXiVyQInLhGCQ2iZQHsp3ZMIrXtsbjKJblD8g/5B+4wpUb4sqFP4Abn0BNZ4KACMFYHtdUd7W7ezopM13ZKPoyF1y5eu36/MKNxZu3bt9Z6t29t1kVtUnVJC2ywmwncaUynauJ1TZT26VR8SzJ1FZytO7sW8fKVLrIX9mmVHuz+DDXU53GltTrXv9ZuDs1cdqeNPujrh11K+fHZn/cteNuZX355DG9okEkK7wMhh704ddG0fuOXRygQIoaMyjksMQZYlR8djBEhJLcHlpyhkiLXaHDIrU1vRQ9YrJH3A952vFszrOLWYk65V8yvobKEI+oKehniN3fQrHXEtmxf4vdSkyXW8Nv4mPNyFq8Ifsv3YXn/+pcLRZTrEoNmjWVwrjqUh+llq64zMNfqrKMUJJz+IB2Q5yK8qLPoWgqqd31Nhb7V/F0rDun3rfGN5+lwrFEbX5m38odatpL6WVDZLnLLXEkhn8OwGWwORoMx4Po5ZP+2nM/HAt4gIdY5gQ8xRpeYAMTZnOKt3iH98FZ8CH4GHw6dw3mvOY+flvB5x9U/qjK</latexit>

(45)

Faites les exercices suivants

p.843 # 1 et 2

(46)

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

(47)

(48)

(49)

Z b a

f (x, c)dx

!

y₁

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

(50)

Z b a

f (x, c)dx

!

y₁

(51)

Z b a

f (x, c)dx

!

y₁

(52)

Z b a

f (x, c)dx

!

y₁

(53)

Z b a

f (x, c)dx

!

y₁

Z d c

f (a, y)dy

!

x₁

<latexit sha1_base64="lfBAmeFh6W4nr3LIMOdr2lDDZEE=">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</latexit>

(54)

Z b a

f (x, c)dx

!

y₁

Z d c

f (a, y)dy

!

x₁

(55)

Z b a

f (x, c)dx

!

y₁

Z d c

f (a, y)dy

!

x₁

(56)

(57)

nX1

k=0

Z b a

f (x, c + k y_k)dx

!

y_k

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

(58)

nX1

k=0

Z d c

f (a + k x_k, y)dy

!

x_k

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

nX1

k=0

Z b a

f (x, c + k y_k)dx

!

y_k

(59)

nX1

k=0

Z d c

f (a + k x_k, y)dy

!

x_k

nX1

k=0

Z b a

f (x, c + k y_k)dx

!

y_k

(60)

nX1

k=0

Z d c

f (a + k x_k, y)dy

!

x_k

nX1

k=0

Z b a

f (x, c + k y_k)dx

!

y_k

(61)

nX1

k=0

Z d c

f (a + k x_k, y)dy

!

x_k

nX1

k=0

Z b a

f (x, c + k y_k)dx

!

y_k

(62)

nX1

k=0

Z d c

f (a + k x_k, y)dy

!

x_k

nX1

k=0

Z b a

f (x, c + k y_k)dx

!

y_k

(63)

nX1

k=0

Z d c

f (a + k x_k, y)dy

!

x_k

nX1

k=0

Z b a

f (x, c + k y_k)dx

!

y_k

(64)

nX1

k=0

Z d c

f (a + k x_k, y)dy

!

x_k

nX1

k=0

Z b a

f (x, c + k y_k)dx

!

y_k

nlim!1

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