Integrales Multiples
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Transcript of Integrales Multiples
![Page 1: Integrales Multiples](https://reader035.fdokument.com/reader035/viewer/2022062217/5695d1b41a28ab9b029796b2/html5/thumbnails/1.jpg)
INTEGRAL TRIPLE
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PROPIEDADES DE LA INTEGRAL TRIPLE
Las Propiedades del 1 al 6 de las integrales dobles se generalizan para las integrales
Triples, en general sobre un sólido Q , se tiene:
donde Q = .
y se llaman «solapamientos»
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dV
CÁLCULO DE INTEGRALES TRIPLES – INTEGRAL ITERADA
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EVALUACIÓN DE INTEGRALES ITERADAS
Si R es el rectángulo R = [a, b] x [c , d] x [u , v ] sobre el cual f es integrable, entonces
![Page 5: Integrales Multiples](https://reader035.fdokument.com/reader035/viewer/2022062217/5695d1b41a28ab9b029796b2/html5/thumbnails/5.jpg)
1. Si R :
La región de integración R ,es proyectada
Sobre el plano XY.
∭𝑅
❑
𝑓 (𝑥 , 𝑦 , 𝑧 )𝑑𝑉=∫𝑎
𝑏 ( ∫∅ 1 (𝑥 )
∅ 2 (𝑥 )
( ∫𝛾1 (𝑥 ,𝑦)
𝛾2 (𝑥 ,𝑦)
𝑓 (𝑥 , 𝑦 ,𝑧 )𝑑𝑧)𝑑𝑦 )𝑑𝑥
REGIONES DE INTEGRACIÓN
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𝐎𝐓𝐑𝐀𝐒𝐑𝐄𝐆𝐈𝐎𝐍𝐄𝐒𝐃𝐄𝐈𝐍𝐓𝐄𝐆𝐑𝐀𝐂𝐈Ó𝐍
X= f(y,z)Y=f(x,z)
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Ejemplo 1
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Proyectando sobre el plano XY, hacemos z = 0 , entonces
y =
x
y
y
x
-2
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∫0
1
∫0
𝑦
∫0
1− 𝑦2
𝑑𝑧𝑑𝑥𝑑𝑦
Determinar el sólido cuyo volumen es dado por la integral
00 x y0 z 1 -
Ejemplo 2
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TEOREMA DE FUBINI PARA INTEGRALES TRIPLES
Si suponemos que la región de integración es de la primera forma Q: a
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Cambio de Variable
,y)
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CAMBIOS DE VARIABLES: JACOBIANOS
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CAMBIO DE VARIABLES EN INTEGRALES TRIPLES
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rcos
rsen
COORDENADAS CILINDRICAS
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CAMBIO A COORDENADAS CILINDRICAS
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DIFERENCIAL DE VOLUMEN EN COORDENADAS CILINDRICAS
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La integral triple en coordenadas cilíndricas
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Coordenadas Esféricas
X=
F(, , )
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CAMBIO A COORDENADAS ESFERICAS
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DIFERENCIAL DE VOLUMEN EN COORDENADAS ESFÉRICAS
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z = 1 -
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y + z = 2 , x = 4 -
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MOMENTOS DE INERCIA DE UNA REGIÓN SÓLIDA
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Cambio de Variable
,y)
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CAMBIOS DE VARIABLES: JACOBIANOS
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CAMBIO DE VARIABLES EN INTEGRALES TRIPLES
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rcos
rsen
COORDENADAS CILINDRICAS
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CAMBIO A COORDENADAS CILINDRICAS
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![Page 38: Integrales Multiples](https://reader035.fdokument.com/reader035/viewer/2022062217/5695d1b41a28ab9b029796b2/html5/thumbnails/38.jpg)
La integral triple en coordenadas cilíndricas
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Coordenadas Esféricas
X=
F(, , )
![Page 40: Integrales Multiples](https://reader035.fdokument.com/reader035/viewer/2022062217/5695d1b41a28ab9b029796b2/html5/thumbnails/40.jpg)
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z = 1 -
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y + z = 2 , x = 4 -
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