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Criado y A. Gallinari" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Objeti vos de la sesi\363n" }}{PARA 15 "" 0 "" {TEXT -1 89 "Utilizar los coma ndos que Maple lleva incorporados para manejar integrales de funcione s." }}{PARA 15 "" 0 "" {TEXT -1 137 "Recordar algunos de los comandos \+ y recursos de programaci\363n de Maple con los que el alumno ya ha ten ido contacto en pr\341cticas anteriores." }}{PARA 15 "" 0 "" {TEXT -1 80 "Repasar y visualizar los conceptos de integral e integral impropia de funciones." }}{PARA 15 "" 0 "" {TEXT -1 123 "Estudiar algunas apli caciones de la integraci\363n al c\341lculo de \341rea de regiones\npl anes y vol\372menes de s\363lidos de revoluci\363n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#presiona return cuando termines de leer" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 261 65 "Comandos de Maple necesarios \+ para la realizaci\363n de esta pr\341ctica" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 98 "Los siguientes son comandos disponibles en Maple n ecesarios para la realizaci\363n de esta pr\341ctica :" }}{PARA 0 "" 0 "" {TEXT -1 8 " " }}{PARA 0 "" 0 "" {TEXT -1 24 " int \+ Int " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 " middlebox middlesum " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 20 " tubeplot " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#p resiona return cuando termines de leer" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Integrales" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 " La in tegral indefinida" }}{PARA 0 "" 0 "" {TEXT -1 100 "El teorema fundamen tal del c\341lculo integral nos permite determinar cu\341ndo el proces o de calcular la " }{TEXT 273 19 "integral indefinida" }{TEXT 259 1 " \+ " }{TEXT -1 7 "(o las " }{TEXT 274 10 "primitivas" }{TEXT -1 18 ") de \+ una funci\363n f" }{TEXT 304 1 " " }{TEXT -1 65 "es el proceso inverso a calcular la derivada de una funci\363n. Si " }{TEXT 283 7 "F'(x) = " }{TEXT -1 0 "" }{TEXT 284 1 " " }{TEXT 305 2 "f " }{TEXT 306 1 " " } {TEXT -1 1 "y" }{TEXT 285 5 " f " }{TEXT 307 13 "es continua, " } {TEXT -1 10 "entonces " }{XPPEDIT 18 0 "F = int(f(x),x);" "6#/%\"FG-% $intG6$-%\"fG6#%\"xGF+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 280 9 "Ejemplo: " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " F:=x^5;" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 10 "calculamos" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:=diff(F,x);\n" }}}{PARA 0 "" 0 "" {TEXT -1 33 "y volvemos a obtener f mediante" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Int(f,x )=int(f,x);" }}}{PARA 0 "" 0 "" {TEXT -1 16 "Obs\351rvese que si" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "G:=x^5+273;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g:=diff(G,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Int(g,x)=int(g,x);" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "Es decir, obtenemos el \+ mismo resultado. " }}{PARA 0 "" 0 "" {TEXT -1 12 "En general, " } {TEXT 275 70 "la integral indefinida s\363lo est\341 definida m\363dul o una constante aditiva" }{TEXT -1 11 ", es decir:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "int(di ff(F(x),x),x) = F(x)+C;" "6#/-%$intG6$-%%diffG6$-%\"FG6#%\"xGF-F-,&-F+ 6#F-\"\"\"%\"CGF1" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 276 11 "Ejercicio: " }{TEXT -1 46 "Calcula la in tegral indefinida de la funci\363n " }{XPPEDIT 18 0 "1/(x^2+x+1)" "6# *&\"\"\"F$,(*$%\"xG\"\"#F$F'F$F$F$!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 62 "Ve rifica este c\341lculo derivando la expresi\363n que has obtenido:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "#presiona return cuando termines de leer" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 " La integral definida" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "La " }{TEXT 277 17 "integral definida" }{TEXT -1 15 " de una funci\363n" }{TEXT 264 1 " " }{TEXT 256 4 "f(x)" }{TEXT -1 23 " no negativa entre los " }{TEXT 279 24 "l\355mites de integraci\363n " }{TEXT 257 3 "x=a" }{TEXT -1 3 " y" }{TEXT 265 1 " " }{TEXT 278 4 " x=b" }{TEXT 266 1 " " }{TEXT -1 66 " es el \341rea encerrada entre la gr\341fica de la funci\363n f (x), el eje " }{TEXT 268 1 "x" }{TEXT -1 15 ", y las rectas " }{TEXT 267 4 "x=a " }{TEXT 260 1 " " }{TEXT -1 3 "y " }{TEXT 269 4 "x=b." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:f:=x->4-x^2;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "with(plots):a:=.5;\nb:=1.5;\np:=plot(f(x),x=0..2,col or=blue):\nq:=plot(\{[a,t,t=0..f(a)],[b,t,t=0..f(b)]\},color=maroon): \ndisplay(\{p,q\});" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 39 "Este \341rea se calcula aproxim\341ndola con " }{TEXT 262 1 "n" }{TEXT 270 1 " " }{TEXT -1 34 "rect\341ngulos y luego increm entando " }{TEXT 263 1 "n" }{TEXT 308 1 " " }{TEXT 271 0 "" }{TEXT -1 15 "sucesivamente: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 217 "n:=7; # Ser\341n n=7 rect\341ngulo s\nh:=(b-a)/n; # h es la anchura de cada rect\341ngulo \np:=plot(f( x),x=0..2):\nq:=plot(f(a+ceil((x-a)/h)*h),x=.5..1.5):\nr:=seq(plot([a+ i*h,t,t=0..f(a+i*h)]),i=0..n):\ndisplay(\{p,q,r\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#presiona return cuando termines de leer " }}}{PARA 0 "" 0 "" {TEXT -1 355 "En el siguiente ejemplo puedes ver \+ gr\341ficamente la forma en que se van obteniendo sucesivas aproximaci ones del \341rea encerrada por f por medio de sumas superiores sobre \+ particiones uniformes. Ejecuta los comandos y haz click sobre el dibuj o que aparece para ver la animaci\363n (el objetivo es ver la animaci \363n, no entender c\363mo hemos empleado los comandos):" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "resta rt;\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f:=x-> 4-x^2;\na:=.5;b:=1.5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 316 "h :=n->(b-a)/n:\npbox:=(x0,h,v)->[[x0,0],[x0,v],[x0+h,v],[x0+h,0]]:\nrec ts:=n->seq(pbox(a+i*h(n),h(n),f(a+i*h(n))),i=0..n-1):\ncurvstruc:=n->C URVES(rects(n),COLOUR(RGB,0.1,0,0)):\nanimstruc:=PLOT(ANIMATE(seq([cur vstruc(n)],n=3..40)), AXESLABELS(\"x\",\"y\"), VIEW(a..b,DEFAULT)):\np :=plot(f(x),x=0..2):\ndisplay(\{p,animstruc\});" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "#presiona return cuando termines de leer" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Vamos aho ra a estimar el \341rea limitada por la gr\341fica de la funci\363n \+ " }{XPPEDIT 18 0 "f(x) = x^2/2+1;" "6#/-%\"fG6#%\"xG,&*&F'\"\"#F*!\"\" \"\"\"F,F," }{TEXT -1 11 " y el eje " }{XPPEDIT 18 0 "OX" "6#%#OXG" } {TEXT -1 21 " entre las abscisas " }{XPPEDIT 18 0 "a=0" "6#/%\"aG\"\" !" }{TEXT -1 6 " y " }{XPPEDIT 18 0 "b=1" "6#/%\"bG\"\"\"" }{TEXT -1 12 " utilizando " }{XPPEDIT 18 0 "n = 3;" "6#/%\"nG\"\"$" }{TEXT -1 61 " rect\341ngulos de la misma longitud y de altura el valor de f (" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 3 ") " }{TEXT 286 18 "en el punto medio " }{TEXT -1 83 "de dichos intervalos. En este caso estam os aproximando la integral de la funci\363n " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 83 " por medio de una suma de Riemann sobr e una partici\363n uniforme del intervalo [0,1]." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "restart:f:=x->(x^2/2)+1;" }}}{PARA 0 "" 0 "" {TEXT -1 219 "Para obtener la representaci\363n gr\341fica de f y de \+ la regi\363n formada por los rect\341ngulos cuyas bases son los subint ervalos resultantes de dividir el intervalo [0,1] en 3 partes iguales y las alturas son los valores de " }{TEXT 309 0 "" }{TEXT -1 63 "f \+ en los puntos medios de dichos subintervalos, utilizamos la " }{TEXT 287 16 "librer\355a student" }{TEXT -1 14 " y el comando " }{TEXT 288 9 "middlebox" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "middl ebox(f(x),x=0..1,3);" }}}{PARA 0 "" 0 "" {TEXT -1 11 "El comando " } {TEXT 310 10 "middlesum " }{TEXT -1 70 "calcula el \341rea de la regi \363n formada por los rect\341ngulos anteriores es" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 36 "middlesum(f(x),x=0..1,3):%=evalf(%);" }}} {PARA 0 "" 0 "" {TEXT -1 208 "Si ahora empleamos 5 rect\341ngulos en l ugar de 3, podemos comprobar gr\341ficamente que la aproximaci\363n de l \341rea encerrada por f es mejor, y obtener el correspondiente valo r num\351rico del \341rea encerrada por ellos:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "middlebox(f(x),x=0..1,5);" }}}{PARA 0 "" 0 "" {TEXT -1 73 "As\355, el \341rea de la regi\363n formada por los cinco \+ rect\341ngulos anteriores es" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "middlesum(f(x),x=0..1,5):%=evalf(%);" }}}{PARA 0 "" 0 "" {TEXT -1 256 "Para representar gr\341ficamente las regiones correspondientes a \+ particiones del intervalo [0,1] en 10, ...,50 rect\341ngulos de las ca racter\355sticas anteriores, podemos observar que la regi\363n limita da por \351stos se asemeja cada vez m\341s a la regi\363n limitada por " }{XPPEDIT 18 0 " f " "6#%\"fG" }{TEXT -1 114 " (ejecuta las sigui entes l\355neas de input de comandos y haz click sobre el dibujo obten ido para ver la animaci\363n):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "for i from 1 to 5 do p[i]:=middlebox(f(x),x=0..1,10*i)od:\nwith (plots):\ndisplay([seq(p[i],i=1..5)],insequence=true);\n" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "As\355 pues, el \+ \341rea de las regiones formadas por esos rect\341ngulos es, en este c aso:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "seq(evalf(middlesum( f(x),x=0..1,10*i)),i=1..5);" }}}{PARA 0 "" 0 "" {TEXT -1 158 "Visto lo anterior es posible determinar una f\363rmula general que proporcione una estimaci\363n del\240\341rea correspondiente a una subdivisi\363n del intervalo [0,1] en " }{XPPEDIT 18 0 "n " "6#%\"nG" }{TEXT -1 47 " rect\341ngulos de las caracter\355sticas anteriores " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "middlesum(f(x),x=0..1,n):%=value(%) ;" }}}{PARA 0 "" 0 "" {TEXT -1 43 "El l\355mite de la expresi\363n ant erior cuando " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 99 " tiende a \+ infinito representa, por el criterio de Riemann, el \341rea encerrada \+ por la gr\341fica de f: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " limit(rhs(%),n=infinity);" }}}{PARA 0 "" 0 "" {TEXT -1 74 "Si ahora ca lculamos directamente la integral de f en el intervalo [0,1]," }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Int(f(x),x=0..1):%=value(%); " }}}{PARA 0 "" 0 "" {TEXT -1 49 "resulta que dicha integral es el l \355mite anterior." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#presi ona return cuando termines de leer" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 " Integraci\363n Impropia" }}{PARA 0 "" 0 "" {TEXT 311 107 "Al e studiar el concepto de integral partimos de dos conceptos fijos: se re quiere que la funci\363n considerada" }{TEXT 312 1 " " }{TEXT 281 83 " est\351 acotada y que el intervalo de integraci\363n sea un intervalo \+ acotado (y cerrado)" }{TEXT -1 117 ". Si alguna de estas dos condicion es no se satisface, hay que introducir ciertos cambios, apareciendo el concepto de " }{TEXT 282 17 "integral impropia" }{TEXT -1 53 " como e l resultado de un proceso de paso al l\355mite. " }}{PARA 0 "" 0 "" {TEXT -1 209 "En cualquier caso, lo primero que hay que observar es qu e Maple V no hace mucha distinci\363n entre integrales propias e impro pias. En este sentido trata el valor infinito como si se tratase de ot ro n\372mero real. " }}{PARA 0 "" 0 "" {TEXT -1 183 "Esto puede servir nos para calcular el valor de una integral impropia: si el c\341lculo del l\355mite obtenido nos proporciona un n\372mero real, diremos que se trata de una integral impropia " }{TEXT 289 11 "convergente" } {TEXT -1 15 "; si el l\355mite " }{TEXT 291 9 "no existe" }{TEXT -1 6 " o es " }{TEXT 290 8 "infinito" }{TEXT -1 29 ", ser\341 una integral \+ impropia " }{TEXT 292 10 "divergente" }{TEXT -1 85 "; si Maple V es in capaz de calcularla, no se puede asegurar si hay convergencia o no." } }{PARA 0 "" 0 "" {TEXT -1 98 "En todo caso, Maple V no estudia la conv ergencia de las integrales impropias, intenta calcularlas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Int( 1/(1+x^2),x=0..+infinity)=\nint(1/(1+x^2),x=0..+infinity);" }}}{PARA 0 "" 0 "" {TEXT -1 76 "Obviamente, la integral impropia anterior conve rge, igual que las siguientes" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Int(x*exp(-x),x=0..+infinity)=\nint(x*exp(-x),x=0..+infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(x*log(x),x=0..1)=int( x*log(x),x=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 31 "Pero las pr\363xima s, no convergen" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int(1/x^2 ,x=0..1)=int(1/x^2,x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Int(cos(x),x=0..infinity)=int(cos(x),x=0..infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#presiona return cuando termines de leer" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Aplicaciones de las integrales" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 24 " \301reas de figuras planas" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Sean f(x) y g(x) dos funciones continuas en un intervalo [a,b]." }}{PARA 0 "" 0 "" {TEXT 302 9 "Ejem plo :" }{TEXT -1 15 " Sean f(x) = " }{XPPEDIT 18 0 "x/sqrt(x^2+1);" "6#*&%\"xG\"\"\"-%%sqrtG6#,&*$F$\"\"#F%F%F%!\"\"" }{TEXT -1 15 " y \+ g(x) = " }{XPPEDIT 18 0 "x^4-x;" "6#,&*$%\"xG\"\"%\"\"\"F%!\"\"" } {TEXT -1 12 " en [-1,2]," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "restart;f:=x->x/sqrt(x^2+1);g:=x->x ^2-x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "with(plots):\na:= plot(\{f(x),g(x)\},x=-1..2,y=-1..2):\nb:=plots[textplot](\{[0.5,0.2,`R `],[0.5,0.6,`f`],\n[-0.8,0.4,`R`],[1.8,1.2,`R`],[1.6,1.6,`g`]\},\nalig n = RIGHT):\ndisplay(\{a,b\});" }}}{PARA 0 "" 0 "" {TEXT -1 114 "Si qu eremos calcular el \341rea de la regi\363n R de la figura anterior, te ndremos que determinar en qu\351 intervalos es " }{XPPEDIT 18 0 "f(x) <= g(x);" "6#1-%\"fG6#%\"xG-%\"gG6#F'" }{TEXT -1 89 ". Los valores ap roximados de los puntos de intersecci\363n de las dos gr\341ficas en [ -1,2] son" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "s:=evalf(solve( f(x)-g(x),x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Por tanto, el \341rea de la regi\363n R ser\341 la suma d e tres integrales:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "I1:=In t(g(x)-f(x),x=-1..s[1])=\nint(g(x)-f(x),x=-1..s[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "I2:=Int(f(x)-g(x),x=s[1]..s[2])=\nint(f(x )-g(x),x=s[1]..s[2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "I2 :=Int(g(x)-f(x),x=s[2]..2)=\nint(g(x)-f(x),x=s[2]..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Area(R):=evalf(int(g(x)-f(x),x=-1.. s[1])+\nint(f(x)-g(x), x=s[1]..s[2])+\nint(g(x)-f(x),x=s[2]..2));" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Al mismo \+ resultado se llega simplemente utilizando la f\363rmula" }}{PARA 257 " " 0 "" {TEXT -1 9 "Area(R)= " }{XPPEDIT 18 0 "int(abs(g(x)-f(x)),x = - 1 .. 2);" "6#-%$intG6$-%$absG6#,&-%\"gG6#%\"xG\"\"\"-%\"fG6#F-!\"\"/F- ;,$F.F2\"\"#" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Int(abs(g(x)-f(x)),x=-1..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#presiona return cuando termines de leer" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 " Vol\372menes de s\363lidos de revoluci\363n" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Los s \363lidos de revoluci\363n se obtienen haciendo girar una regi\363n pl ana alrededor de los ejes de coordenadas. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 293 18 "M\351todo de discos: " }{TEXT -1 5 " Sea " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 60 " una funci\363n continua y no negativ a en un intervalo [a,b]. " }}{PARA 0 "" 0 "" {TEXT -1 65 "Al girar el \+ trapecio curvil\355neo R (determinado por la gr\341fica de " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 111 ", las rectas x= a, x=b y el eje de las x) alrededor del propio eje de las x, se obtien e un s\363lido de revoluci\363n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 297 11 "Ejemplo 1: " }{TEXT -1 12 "Sea f(x) = \+ " }{XPPEDIT 18 0 "2*x^2+3;" "6#,&*&\"\"#\"\"\"*$%\"xGF%F&F&\"\"$F&" } {TEXT -1 27 " en el intervalo [0,1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;f:=x->2*x^2+3 ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "with(plots):a:=0;\nb: =1;\np:=plot(f(x),x=-0.5..1.5,color=blue):\nq:=plot(\{[a,t,t=0..f(a)], [b,t,t=0..f(b)]\},color=maroon):\nl:= plots[textplot]([0.5,2,`R`],alig n = RIGHT):\ndisplay(\{p,q,l\});" }}}{PARA 0 "" 0 "" {TEXT -1 115 "En \+ nuestro caso, el s\363lido de revoluci\363n obtenido al girar el trape cio curvil\355neo R (determinado por la gr\341fica de " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 119 ", las rectas x=0, x=1 y el \+ eje de las x) alrededor del propio eje de las x est\341 delimitado por la siguiente superficie:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "with(plots):with(plots,tubeplot):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "s:=tubeplot([x,0,0],x=0.. 1,radius=f(x),tubepoints=20): display(s,axes=normal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#presiona return cuando termines de leer" }}}{PARA 0 "" 0 "" {TEXT -1 3 "El " }{TEXT 294 34 "volumen de u n s\363lido de revoluci\363n" }{TEXT -1 1 " " }{TEXT 295 26 "alrededor del eje de las x" }{TEXT -1 28 " viene dado por la f\363rmula " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "V = Int(Pi*f(x)^2,x = a .. b);" "6#/%\"VG-%$IntG6$*&%#P iG\"\"\"*$-%\"fG6#%\"xG\"\"#F*/F/;%\"aG%\"bG" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "donde el produc to A(x)= " }{XPPEDIT 18 0 "Pi*f(x)^2;" "6#*&%#PiG\"\"\"*$-%\"fG6#%\"xG \"\"#F%" }{TEXT -1 129 " representa el \341rea de la secci\363n circul ar obtenida cortando el s\363lido de revoluci\363n por un plano perpen dicular al eje de las x. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 19 "En nuestro ejemplo," }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "V:=Int(Pi*f(x)^2,x = 0 .. 1)=int(Pi*(f(x)^2),x=0..1); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#presiona return cuando termines de leer" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 66 "Si ahora queremos determinar el volumen de un s\363lido de revoluci\363n" }{TEXT 298 1 " " }{TEXT -1 26 "alrededor del eje de las x" }{TEXT 299 1 " " }{TEXT -1 77 "determinado por una regi\363n p lana limitada por la gr\341ficas de dos funciones " }{XPPEDIT 18 0 " f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " y " }{XPPEDIT 18 0 "g(x);" "6# -%\"gG6#%\"xG" }{TEXT -1 69 ", primero tenemos que determinar en que \+ intervalos se verifica que " }{XPPEDIT 18 0 "f(x) <= g(x);" "6#1-%\"f G6#%\"xG-%\"gG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 296 10 "Ej emplo 2:" }{TEXT -1 15 " Sean f(x) = " }{XPPEDIT 18 0 "2*x^2+3;" "6# ,&*&\"\"#\"\"\"*$%\"xGF%F&F&\"\"$F&" }{TEXT -1 47 " y g(x) = -2x \+ +7 en el intervalo (0,2)," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g:=x->-2*x+7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "with(plots):a:=0;\nb:=2;\np:=plot( \{f(x),g(x)\},x=0..2.5):\nq:=plot(\{[b,t,t=0..f(b)]\},color=maroon):\n k:= plots[textplot](\{[0.3,5,`R`],[1.7,5,`R`]\},\nalign = RIGHT):\ndis play(\{p,q,k\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(f (x)-g(x),x);" }}}{PARA 0 "" 0 "" {TEXT -1 237 "Podemos dibujar la regi \363n obtenida del siguiente modo (la regi\363n es el s\363lido compre ndido entre las dos superficies dibujadas, haz click sobre el dibujo q ue aparece y utiliza el rat\363n para ver distintas perspectivas del s \363lido obtenido):" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "with (plots):with(plots,tubeplot):\np1:=tubeplot([x,0,0],x=0..1,radius=g(x) ,tubepoints=20): q1:=tubeplot([x,0,0],x=0..1,radius=f(x),tubepoints=20 ): p2:=tubeplot([x,0,0],x=1..2,radius=f(x),tubepoints=20):\nq2:=tubepl ot([x,0,0],x=1..2,radius=g(x),tubepoints=20):\ndisplay(\{p1,p2,q1,q2\} ,axes=normal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#presiona return cuando termines de leer" }}}{PARA 0 "" 0 "" {TEXT -1 32 "En nu estro ejemplo con f(x) = " }{XPPEDIT 18 0 "2*x^2+3;" "6#,&*&\"\"#\" \"\"*$%\"xGF%F&F&\"\"$F&" }{TEXT -1 223 " y g(x) = -2x+7 en e l intervalo (0,2), el volumen del solido de revoluci\363n obtenido ro tando alrededor del eje de las x la regi\363n limitada por y = f(x), y = g(x), x = 0 y x = 2 ser\341 la suma de dos vol\372menes:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "V:=Int(Pi*(g(x)^2-f(x)^2),x=0..1)+\nInt(Pi*(g(x)^2-f(x)^2),x=1..2 )=\nint(Pi*(g(x)^2-f(x)^2),x=0..1)+\nint(Pi*(g(x)^2-f(x)^2),x=0..1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 300 17 "M\351 todo de capas: " }{TEXT -1 5 "Sea " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6 #%\"xG" }{TEXT -1 126 " una funci\363n continua y no negativa en un i ntervalo [a,b]. Al girar el trapecio curvil\355neo R (determinado por la gr\341fica de " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 103 ", las rectas x=a, x=b y el eje de las x) alrededor del eje de \+ las y se obtiene un s\363lido de revoluci\363n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 11 "Ejemplo 3: " }{TEXT -1 17 "Sea, como antes, " }{TEXT 303 1 " " }{TEXT -1 8 "f(x) = 2" } {XPPEDIT 18 0 "x^2+3;" "6#,&*$%\"xG\"\"#\"\"\"\"\"$F'" }{TEXT -1 12 " \+ en [0,1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 217 "restart;with(plots):\nf:=x->2*x^2+3;\nu:=plot(f(x) ,x=0..1,color=blue):\nv:=plot(\{[0.5,t,t=0..f(0.5)],[0.2,t,t=0..f(0.2) ],\n[1,t,t=0..f(1)]\},color=maroon):\nw:= plots[textplot]([0.35,2,`A`] ,align = RIGHT):\ndisplay(\{u,v,w\});" }}}{PARA 0 "" 0 "" {TEXT -1 104 "En el dibujo anterior se representa la porci\363n A de la regi \363n correspondiente al intervalo [0.2,0.3]. " }}{PARA 0 "" 0 "" {TEXT -1 139 "La siguiente figura representa la porci\363n de volumen \+ de rotaci\363n alrededor del eje y determinada por A, cuyo valor se pu ede aproximar por " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" } {TEXT -1 15 " (0.4) f(0.4): " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "with(plots):with(plots,tube plot):\nc:=plot3d(0.2,theta=0..2*Pi,z=0..f(0.2),\nstyle=patch,coords=c ylindrical):\nd:=plot3d(0.5,theta=0..2*Pi,z=0..f(0.5),\nstyle=patch,co ords=cylindrical):\ndisplay(\{c,d\},axes=normal);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Entonces el volumen del s \363lido generado por la regi\363n R es igual a" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "Volumen = Int(2*pi*x*f(x),x = a .. b);" "6#/%(VolumenG- %$IntG6$**\"\"#\"\"\"%#piGF*%\"xGF*-%\"fG6#F,F*/F,;%\"aG%\"bG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "V:=Int(2*pi*x*f(x),x=0..1)=int(2*pi*x*f(x),x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "#fin " }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 28 "Cuestiones sobre integraci\363n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 313 4 " " }}{PARA 0 "" 0 "" {TEXT -1 64 " 1. Estudia la convergencia o la divergencia de la integral " } {XPPEDIT 18 0 "Int(1/(exp(1)^x+15),x = 0 .. infinity);" "6#-%$IntG6$*& \"\"\"F',&)-%$expG6#F'%\"xGF'\"#:F'!\"\"/F-;\"\"!%)infinityG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "a) La integral diverge" }} {PARA 0 "" 0 "" {TEXT -1 29 "b) La integral converge a " }{XPPEDIT 18 0 "1/15*ln(exp(1)+15)-1/15;" "6#,&*(\"\"\"F%\"#:!\"\"-%#lnG6#,&-%$e xpG6#F%F%F&F%F%F%*&F%F%F&F'F'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "c) La integral converge a " }{XPPEDIT 18 0 "4/15*ln(2);" "6 #*(\"\"%\"\"\"\"#:!\"\"-%#lnG6#\"\"#F%" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 63 "2. Estudia la convergencia o la diver gencia de la integral " }{XPPEDIT 18 0 "Int(1/sqrt(x^4-1),x = 1 .. \+ 3);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*$%\"xG\"\"%F'F'!\"\"F//F-;F'\" \"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 30 "a) La integral c onverge a " }{XPPEDIT 18 0 ".9772817885;" "6#-%&FloatG6$\"+&)y\"Gx*! #5" }}{PARA 0 "" 0 "" {TEXT -1 29 "b) La integral converge a " } {XPPEDIT 18 0 "1/15*ln(exp(1)+15)-1/15;" "6#,&*(\"\"\"F%\"#:!\"\"-%#ln G6#,&-%$expG6#F%F%F&F%F%F%*&F%F%F&F'F'" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "c) La integral converge a " }{XPPEDIT 18 0 ".3473 148625;" "6#-%&FloatG6$\"+D'[JZ$!#5" }{TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "3. Sea F( x) = " }{XPPEDIT 18 0 "Int(exp(t^2),t = 0 .. x)^2" "6#*$-%$IntG6$-%$e xpG6#*$%\"tG\"\"#/F+;\"\"!%\"xGF," }{TEXT -1 15 " y sea G(x) = " } {TEXT 314 0 "" }{XPPEDIT 18 0 "exp(x^2);" "6#-%$expG6#*$%\"xG\"\"#" } {TEXT -1 26 ". Se puede verificar que " }{XPPEDIT 18 0 "Limit(F(x),x \+ = infinity);" "6#-%&LimitG6$-%\"FG6#%\"xG/F)%)infinityG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Limit(G(x),x = infinity);" "6#-%&LimitG6$-%\"GG6# %\"xG/F)%)infinityG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "infinity;" "6#% )infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "Utilizando \+ la regla de L'H\364pital para calcular " }{XPPEDIT 18 0 "Limit(F(x)/G (x),x = infinity);" "6#-%&LimitG6$*&-%\"FG6#%\"xG\"\"\"-%\"GG6#F*!\"\" /F*%)infinityG" }{TEXT -1 42 " se verifica que este l\355mite es ig ual a" }}{PARA 0 "" 0 "" {TEXT -1 6 "a) 3" }}{PARA 0 "" 0 "" {TEXT -1 6 "b) 0" }}{PARA 0 "" 0 "" {TEXT -1 14 "c) no existe" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 18 "4. Sean f(x) = " }{XPPEDIT 18 0 "x^2-6;" "6#,&*$%\"xG\"\"#\" \"\"\"\"'!\"\"" }{TEXT -1 15 " y g(x) = " }{XPPEDIT 18 0 "12-x^2; " "6#,&\"#7\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 16 " en [-5,5]." }} {PARA 0 "" 0 "" {TEXT -1 77 "El \341rea de la regi\363n limitada por \+ y = f(x), y = g(x), x = -5 y x =5 es:" }}{PARA 0 "" 0 "" {TEXT -1 6 "a) " }{XPPEDIT 18 0 "392/3;" "6#*&\"$#R\"\"\"\"\"$!\"\"" }} {PARA 0 "" 0 "" {TEXT -1 4 "b) " }{XPPEDIT 18 0 "-392/3;" "6#,$*&\"$# R\"\"\"\"\"$!\"\"F(" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "c) \+ " }{XPPEDIT 18 0 "376/3;" "6#*&\"$w$\"\"\"\"\"$!\"\"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 60 " 5. Hallar el \341rea encerrada por la gr\341fica de la funci \363n " }{XPPEDIT 18 0 "f(x) = abs(x^2-4*x+3);" "6#/-%\"fG6#%\"xG-%$a bsG6#,(*$F'\"\"#\"\"\"*&\"\"%F.F'F.!\"\"\"\"$F." }{TEXT -1 18 " entre x=0 y x=4." }}{PARA 0 "" 0 "" {TEXT -1 5 "a) 3" }}{PARA 0 "" 0 "" {TEXT -1 5 "b) 4" }}{PARA 0 "" 0 "" {TEXT -1 6 "c) -2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{PARA 0 "" 0 "" {TEXT -1 58 "6. Hallar el \341rea encerrada por l a gr\341fica de la funci\363n " }{XPPEDIT 18 0 "f(x) = 1/(1+x^2);" "6 #/-%\"fG6#%\"xG*&\"\"\"F),&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 16 " y su \+ as\355ntota." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 20 "7. Sean f(x) = " }{XPPEDIT 18 0 "x^2-6;" "6#,&*$%\"xG\"\"# \"\"\"\"\"'!\"\"" }{TEXT -1 15 " y g(x) = " }{XPPEDIT 18 0 "12-x^ 2;" "6#,&\"#7\"\"\"*$%\"xG\"\"#!\"\"" }{TEXT -1 35 " en [-5,5] . \+ (Ver Problema 4)" }}{PARA 0 "" 0 "" {TEXT -1 105 "El volumen del soli do de revoluci\363n obtenido rotando alrededor del eje de las x la reg i\363n limitada por " }}{PARA 0 "" 0 "" {TEXT -1 44 " y = f(x), y = g(x), x = -5 y x =5 es:" }}{PARA 0 "" 0 "" {TEXT -1 5 "a) " } {XPPEDIT 18 0 "784*Pi;" "6#*&\"$%y\"\"\"%#PiGF%" }}{PARA 0 "" 0 "" {TEXT -1 4 "b) " }{XPPEDIT 18 0 "-784*Pi;" "6#,$*&\"$%y\"\"\"%#PiGF&! \"\"" }}{PARA 0 "" 0 "" {TEXT -1 4 "c) " }{XPPEDIT 18 0 "608*Pi;" "6# *&\"$3'\"\"\"%#PiGF%" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 99 "8. Sea C l a circunferencia de radio 2 con centro en el punto (3,0). La ecuaci \363n que define C es " }{XPPEDIT 18 0 "(x-3)^2+y^2 = 4;" "6#/,&*$,&% \"xG\"\"\"\"\"$!\"\"\"\"#F(*$%\"yGF+F(\"\"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 109 "El volumen del solido de revoluci\363n obtenid o rotando alrededor del eje de las y la regi\363n limitada por C es: " }}{PARA 0 "" 0 "" {TEXT -1 5 "a) " }{XPPEDIT 18 0 "24*Pi^2;" "6#*& \"#C\"\"\"*$%#PiG\"\"#F%" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "b) " }{XPPEDIT 18 0 "12*Pi^2;" "6#*&\"#7\"\"\"*$%#PiG\"\"#F%" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "c) " }{XPPEDIT 18 0 "6*P i^2;" "6#*&\"\"'\"\"\"*$%#PiG\"\"#F%" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 65 "9. Halla el \+ valor de la constante C tal que la integral impropia " }{XPPEDIT 18 0 "Int(x/(x^2+1)-C/(3*x+1),x = 0 .. infinity);" "6#-%$IntG6$,&*&%\"xG\" \"\",&*$F(\"\"#F)F)F)!\"\"F)*&%\"CGF),&*&\"\"$F)F(F)F)F)F)F-F-/F(;\"\" !%)infinityG" }{TEXT -1 11 " converge." }}{PARA 0 "" 0 "" {TEXT -1 45 "a) la integral diverge para todo valor de C" }}{PARA 0 "" 0 "" {TEXT -1 4 "b) " }{XPPEDIT 18 0 "C = 0;" "6#/%\"CG\"\"!" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "c) " }{XPPEDIT 18 0 "C = 3;" "6#/%\" CG\"\"$" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{PARA 0 "" 0 "" {TEXT -1 40 "10. Verifica que la integral i mpropia " }{XPPEDIT 18 0 "Int(x,x = -infinity .. infinity);" "6#-%$In tG6$%\"xG/F&;,$%)infinityG!\"\"F*" }{TEXT -1 25 " diverge. Ahora calc ula " }{XPPEDIT 18 0 "Limit(Int(x,x = -c .. c),c = infinity);" "6#-%&L imitG6$-%$IntG6$%\"xG/F);,$%\"cG!\"\"F-/F-%)infinityG" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 23 " Se verifica que:" }}{PARA 0 " " 0 "" {TEXT -1 6 "a) " }{XPPEDIT 18 0 "Limit(Int(x,x = -c .. c),c \+ = infinity)" "6#-%&LimitG6$-%$IntG6$%\"xG/F);,$%\"cG!\"\"F-/F-%)infini tyG" }{TEXT -1 9 " diverge" }}{PARA 0 "" 0 "" {TEXT -1 6 "b) " } {XPPEDIT 18 0 "Limit(Int(x,x = -c .. c),c = infinity) = 1/2;" "6#/-%&L imitG6$-%$IntG6$%\"xG/F*;,$%\"cG!\"\"F./F.%)infinityG*&\"\"\"F3\"\"#F/ " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "c) " }{XPPEDIT 18 0 "Limit(Int(x,x = -c .. c),c = infinity) = 0;" "6#/-%&LimitG6$-%$IntG6$ %\"xG/F*;,$%\"cG!\"\"F./F.%)infinityG\"\"!" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}}{MARK "1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }