integrales dobles
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integrales dobles...
Description
“Año de la Consolidación Económica y Social del Perú”
CARRERA PR"+ESI"&A# DE I&'E&IERIA CIVI#
Interales Do!les CATEDRA: Cálculo Vectorial
CATEDRATIC": #ic$ %ore #uis "rtea Varas I&TE'RA&TES: • • • • •
(E&D")A RA("S* Anel (anuel RA("S C"RAS(A* +ranco &ider TI&E" R,A* Ilde CAR-A%A# ',I##E&* S.irley$ CARTA'E&A CARTA'E&A PARI* Dieo Die o A$
Ciclo: III
Huancavelica - Perú 2011
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Cálculo Vectorial
Integrales Dobles
DEDICATORIA A nuestros padres, por todo el apoyo que nos brindan día a día para para seguir seguir cumpl cumplien iendo do nuestr nuestras as metas.
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Cálculo Vectorial
Integrales Dobles
INTRODUCCIÓN Las matemáticas en el mundo real, son bastante más complicadas que los problemas comunes que se presentan día a día. Las integrale integraless se usan en el cálculo infinites infinitesimal imal y para calcular calcular areas, areas, utilizamos integrales dobles. Las integrales son sumas, el símbolo integral viene de ahí, luego el objetivo de integrales de superficie es sumar superficies infinitesimales y las de volumen, sumar volúmenes infinitesimales. Las coordenadas son otra cosa a parte, se usan porque no todo se puede epres resar
rápido
en
un
sist istema
de
coordenadas
cartesian iano
!,y, ,y,z".
#or ejemplo, si tienes un cilíndro o un cono y quieres hallar su área, por poder, lo puedes hacer en cartesianas, pero hay herramientas herramientas matemáticas mejores, mejores, como epresarlo en coordenadas cilíndricas e integrar. $s por facilitar los cálculos, si un problema tiene simetría esf%rica, se usan coordenadas esf%ricas !electromagnetismo", si tiene simetría parab&lica, se usan coordenadas parab&licas !astrofísica", si usamos c&nicas, suele ser más fácil hacerlo en polares etc. $n definitiva, que según que, hay figuras que son más fáciles de definir matemática matemáticamente mente en funci&n funci&n del ángulos ángulos y otras magnitudes magnitudes que en funci&n de coordenadas cartesianas. 'e todos modos, estudia porque esto es básico, las coordenadas polares, esf%ricas y cilíndricas cilíndricas son básicas. A continuaci&n detallamos detallamos todo lo referido a integrales 'obles. 'obles. Los alumnos
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Cálculo Vectorial
Integrales Dobles
I&TE'RA#ES D"-#ES
#($)($*+-/-0 #ara la comprensi&n adecuada de este capítulo de las integrales múltiples se requiere del conocimiento previo de0 1%todos de integraci&n -uperficies 2eometría analítica 3oordenadas polares /45$6/-0 $stablecer los fundamentos necesarios para la interpretaci&n y aplicaci&n de la integral doble, al finalizar este capítulo el alumno debe estar en capacidad de utilizar la integral doble en el cálculo de áreas, volumen, centro de masas, etc. así como tambi%n el cálculo de coordenadas dobles y emplear jacobianos. 7(/'+33/7.
$n
el
estudio
de
las
integrales
ordinarias
LA 7$2(AL '/4L$ -/4($ +7 ($3A72+L/0 3onsideremos una funci&n f 'efinida en el rectángulo.
d R c
0
a
b
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Cálculo Vectorial
Integrales Dobles
La participaci&n # del rectángulo (, descompone al rectángulo ( en m n rectángulos rectángulos es decir 0
d=yn
Rij Yj-1
C=Y0
X i-
0
a=X
Xi
b=X
$n cada rectángulo ( ij, la funci&n f toma un valor máimo 1 ij, y un valor mínimo mij, Luego se tiene0
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Cálculo Vectorial
Integrales Dobles
*ue reciben los nombres de suma superior p de f y se denota por0
8 suma inferior p de f y se denota por0
$n forma similar del caso de las funciones de una variable se tiene 0 -i f es una funci&n continúa, eiste un numero que satisface la desigualdad.
y R !
' 9
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Cálculo Vectorial
Integrales Dobles
La norma de participaci&n p representada por # se define como la longitud de la diagonal mayor de los rectángulos contenidos en '.
2eom%tricamente la suma de (iemann represente el volumen aproimado del solido bajo la superficie z:f!,y" y que tiene como base la regi&n cerrada '.
# %&'y(!%&'y(
$=!%&'y(
0 y %&'y(
&
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Cálculo Vectorial
Integrales Dobles
#(/#$'A'$- ;+7'A1$7AL$- '$ 7$2(AL '/4L$0 .
?.
@.
.
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Cálculo Vectorial
Integrales Dobles
.
0
&
Luego el punto crítico es p !9,9" ahora calculamos los puntos críticos en el borde como0
Luego el valor mínimo es f !9,9":C y el valores máimo es f!9,D=":=@ de acuerdo a la propiedad @ se tiene
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Cálculo Vectorial
Integrales Dobles
%3'-2(
2raficando la regi&n ' 3alculando los puntos críticos en el interior de la regi&n.
INTEGRALES MÚLTIPLES E INTEGRALES ITERADAS
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Cálculo Vectorial
Integrales Dobles
es la integral con respecto a x de de la funci&n de x 0
+na integral doble, en cambio está definida con respecto a un área en el plano y. La integral doble eiste si y s&lo si las dos integrales iteradas eisten y son iguales. $s decir, si la integral doble eiste, entonces es igual a la integral iterada, sin importar si el orden de integraci&n es dydx & dxdy, y por lo general uno la calcula calculando una sola de estas. -in embargo, a veces las dos integrales iteradas eisten sin ser iguales y en este caso no eiste la integral doble, ya que se tiene0
'e una manera más formal, el eorema de ;ubini afirma ;ubini afirma que
$sto es, si la integral es absolutamente convergente, entonces entonces la integral doble es igual a la integral iterada.
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Cálculo Vectorial
Integrales Dobles
se puede usar si se desea ser enfático al referirse a una integral doble y no a una iterada.
MÉTODOS DE INTEGRACIÓN 1. Func Funcio ione ne con! con!"n "n!e !e $n el caso de funciones constantes, el resultado es trivial0 simplemente multiplíquese el valor de la funci&n constante c por la medida del dominio de integraci&n. -i c : da el volumen de la regi&n y así sucesivamente. #or ejemplo0 y ntegrando f sobre sobre D0
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Cálculo Vectorial
Integrales Dobles
8a que tanto = sin! x " como >y > son funciones impares, y eiste simetría tanto con respecto al eje como con respecto al eje y, las primeras dos inte integr gral ales es se nuli nulific fican an,, de tal tal form forma a que que la inte integr gral al orig origin inal al es igua iguall únicamente a la tercera.
'REA POR DO(LE INTEGRACIÓN La aplicaci&n más simple de las integrales dobles es para hallar el área de una regi&n del plano xy. $sta área está dada por una cualquiera de las integrales
!E" Los límites de integraci&n apropiados. 8a hemos visto como se hace esto en la
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Cálculo Vectorial
Integrales Dobles
#ara interpretar
la
primera integraci&n respecto a x, como suma de todos los elementos != xy
-ituados en una faja horizontal que se etiende desde la curva x=g 1(y) a izquierda hasta la curva x=g 2 2(y) ( y) a la derecha. $l cálculo de esta integral es
$sta $sta última última integr integral al podía podía habers haberse e escrit escrito o de primer primera a intenc intenci&n i&n,, puesto puesto que epresa el área como límite de la suma de fajas horizontales.
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Cálculo Vectorial
Integrales Dobles
ntegrando la funci&n transformada en el dominio de integraci&n correspondiente a las variables , y multiplicando por el valor absoluto del determinante jacobiano y por la serie de diferenciales, se obtiene una integral múltiple que es igual a la integral original, si es que esta eiste.
A continuaci&n continuaci&n se dan algunos algunos ejemplos de estas estas transformaciones. Coordenadas Polares
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Cálculo Vectorial
Integrales Dobles
#or ejemplo0 -i la funci&n es aplicando la transformaci&n se obtiene la funci&n fácilmente integrable con respecto a ϕy a J.
-e pueden obtener funciones incluso más simples0 -i la funci&n es +no tiene0
-i aplica la identidad trigonom%trica pitag&rica trigonom%trica pitag&rica de senos y cosenos. $l determinante jacobiano de la transformaci&n es0
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Cálculo Vectorial
Integrales Dobles
0
razando razando rectas a trav%s del polo y círculos con centro en el polo, se obtiene una partici&n # de la regi&n ', que viene a ser una red de NnO regiones llamadas rectangulares curvea curveados.
/
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Cálculo Vectorial
Integrales Dobles
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Cálculo Vectorial
Luego la integral en coordenadas polares es0
Integrales Dobles
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Cálculo Vectorial
Integrales Dobles
a"
v
.
Y ,
,
u
X
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Cálculo Vectorial
Integrales Dobles
CAM(IO DE )ARIA(LES EN LAS INTEGRALES DO(LES $n las las inte integr gral ales es ordi ordina nari rias as el m%to m%todo do de sust sustitituc uci& i&n n nos nos permi permití tía a calc calcul ular ar integrales complicadas, transformándola en otras más sencillas, es decir0
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Cálculo Vectorial
Integrales Dobles
*ue hace corresponder a un punto !u,v" del plano uv, el punto imagen !,y" del plano P8 y que la aplicaci&n puede epresarse mediante una funci&n vectorial. $n el plano trazamos el radio vector de la regi&n regi&n ', ', el vector vector
que une el origen !9,9" con el punto !,y"
depende depende de u y v, v, y se se puede puede conside considerar rar como como una
funci&n vectorial de dos variables definida por la ecuaci&n0
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Cálculo Vectorial
Integrales Dobles
Luego el centro de la masa de la lámina es el puno #!P,8" donde0
F =do. 1/1$7/ '$ 7$(3A '$ +7A LQ17A.) 3onsideremos una partícula de masa m que se encuentra a una distancia d
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Cálculo Vectorial
Integrales Dobles
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Cálculo Vectorial
:
Integrales Dobles
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Cálculo Vectorial
Integrales Dobles
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Cálculo Vectorial
Integrales Dobles
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Cálculo Vectorial
Integrales Dobles
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Cálculo Vectorial
Integrales Dobles
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Cálculo Vectorial
Integrales Dobles
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