Cálculo integral
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Dirección y realización del proyecto
LCC. Gabriel Barragán Casares Director general del Colegio de Bachilleres del estado de Yucatán
Cálculo integral ISBN: 978-607-489-304-5
Planeación y coordinación
Lic. Alejandro Salazar Ortega Director académico
Metodología y estrategia didáctica
Lic. Lorenzo Escalante Pérez Jefe del Departamento de Servicios Académicos
Coordinación
L.M. Davy Alejandro Pérez Chan
DERECHOS RESERV RESERVADOS ADOS
Queda prohibida la reproducción o trans-
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La reforma integral de la Educación Media Superior La Educación Media Superior (EMS) en México enfrenta desafíos que podrán ser
atendidos sólo si este nivel educativo se desarrolla con una identidad denida que permita a sus distintos actores avanzar ordenadamente hacia los objetivos propuestos. Es importante saber que la EMS en el país está compuesta por una serie de subsistemas que operan de manera independiente, sin correspondencia a un
panorama general articulado articulado y sin que exista suciente comunicación entre ellos. El reto es encontrar los objetivos comunes de esos subsistemas para potenciar sus alcances y de esta manera lograr entre todos, reglas claras de operación. Es importante para el desarrollo de la EMS, que ustedes docentes y estudiantes conozcan los ejes que la regulan, cómo opera y los retos que enfrenta en la actualidad para asumir a partir de dicho conocimiento una actitud diferente que nos permita coadyuvar en este esfuerzo. Los diferentes subsistemas de la EMS han realizado cambios en sus estruc-
turas los cuales pretendieron dar la pertinencia, ecacia y calidad necesarias para que la población a la que atiende (jóvenes entre los 15 y 21 años aproximadamente) adquiriera conocimientos y habilidades que les permitan desarrollarse de manera satisfactoria, ya sea en sus estudios superiores o en el trabajo y, de manera más general, en la vida. En esta misma línea, no se debe perder de vista el contexto social de la EMS: de ella egresan individuos en edad de d e ejercer sus derechos y obligaciones como ciudadanos, y cómo tales deben reunir, en adición a los conocimientos y ha-
bilidades que denirán su desarrollo personal, una serie de actitudes y valores que tengan un impacto positivo en su comunidad y en el país en su conjunto. Es en este contexto que las autoridades educativas del país han propuesto la Reforma Integral de la Educación Media Superior (RIEMS), cuyos objetivos consisten en dar identidad, calidad, equidad y pertinencia a la EMS, a través de mecanismos que permitan articular los diferentes actores en un Sistema Nacional de Bachillerato, dentro del cual se pueda garantizar además de lo anterior, tránsito de estudiantes,
intercambio de experiencias de aprendizaje y la certicación de los mismos. Lo anterior será posible a partir del denominado Marco Curricular Común (MCC) de la RIEMS, el cual se desarrolla considerando el modelo de competencias, y que incluye: Competencias Genéricas, Competencias Disciplinares (básicas y extendidas) y Competencias Profesionales (básicas y extendidas). Esta estructura permite observar de manera clara, los componentes comunes entre los diversos subsistema s, así como aquellos que son propios de cada uno y que por consiguiente, los hace distintos. Lo anterior muestra cómo la RIEMS respeta la diversidad del nivel educativo del país, pero hace posible el Sistema Nacional del Bachillerato, conformado por las distintas instituciones y subsistemas que operan en nuestro país. Una competencia es la integración de habilidades, conocimientos y acti-
tudes en un contexto especíco. Esta estructura reordena y enriquece los planes y programas de estudio existentes y se adapta a sus objetivos; no busca reempla-
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Nuestro subsistema pertenece al conjunto de los que ofrecen bachillerato general, el cual en la denición del MCC de la reforma integral, deberá de sarrollar en los estudiantes capacidades que les permitan adquirir competencias genéricas, competencias disciplinares básicas y extendidas, además de comp etencias profesionales básicas. Las competencias genéricas son las que todos los bachilleres deben estar
en capacidad de desempeñar; las que les permiten comprender el mundo e inuir en él; les capacitan para continuar cont inuar aprendiendo de forma autónoma autónom a a lo largo de sus vidas, y para desarrollar relaciones armónicas con quienes les rodean, así como parti-
cipar ecazmente en los ámbitos social, profesional y político. Dada su importancia, dichas competencias se identican también como competencias clave y constituyen el perl del egresado del Sistema Nacional de Bachillerato. A continuación se listan las competencias genéricas, agrupadas en sus categorías correspondientes:
Se autodetermina y cuida de sí
1.
Se conoce y valora a sí mismo y aborda problemas y retos teniendo en cuenta los objetivos que persigue.
2.
Es sensible al arte y participa en la apreciación e interpretación de sus expresiones en distintos géneros. Elige y practica estilos de vida saludables.
3.
Se expresa y comunica
4.
Escucha, interpreta y emite mensajes pertinentes en distintos contextos mediante la utilización de medios, códigos y herramientas he rramientas apropiados. apropiados.
Piensa crítica y reexivamente
5.
Desarrolla innovaciones innovaciones y propone soluciones a problemas a partir de métodos establecidos.
6.
Sustenta una postura personal sobre temas de interés y relevancia general, considerando otros puntos de vista de manera crítica y reexiva.
Aprende de forma autónoma
7.
Aprende por iniciativa e interés propio a lo largo de la vida.
Trabaja en forma colaborativa
8.
Participa y colabora de manera efectiva en equipos diversos.
Participa con responsabilidad en la sociedad
9.
Participa con una conciencia cívica y ética en la vida de su comunidad, región, México y el mundo.
10. Mantiene una actitud respetuosa hacia la interculturalidad y la diversidad de creencias, valores, ideas y prácticas sociales. 11. Contribuye al desarrollo sustentable de manera crítica, con acciones responsables.
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Las competencias disciplinares básicas procuran expresar las capacidades que todos los estudiantes deben adquirir, adquirir, independientemente del plan y programas de estudio que cursen y la trayectoria académica o laboral que elijan al terminar sus estudios de bachillerato. Las competencias disciplinares básicas dan sustento a la formación de los estudiantes en las comp etencias genéricas que integran el perl de egreso de la EMS y pueden aplicarse en distintos enfoques educati vos, contenidos y estructuras curriculares; se organizan en los c ampos disciplinares siguientes: Matemáticas, Ciencias Experimentales (Física, Química, Biología y Ecología), Ciencias Sociales y Humanidades (Historia, Sociología, Política, Economía, Administración, Administrac ión, Lógica, Ética, Filosofía y Estética) y Comunicación (Lectura y Ex presión oral y escrita, Literatura, Lengua extranjera e Informática). Las competencias disciplinares extendidas dan sustento a las competen-
cias genéricas del perl del egresado del bachillerato, además de que tienen como propósito preparar al estudiante para el nivel superior de estudios, especicando en los elementos disciplinares correspondientes y en su caso, incrementando la
complejidad de la competencia a desarrollar. Al igual que las disciplinares básicas se agrupan en los campos de conocimiento del Bachillerato General.
Matemáticas 1.
Construye e interpreta modelos matemáticos mediante la aplicación de procedimientos aritméticos, algebraicos, geométricos y variacionales, para la comprensión y análisis de situaciones reales, hipotéticas o formales.
2.
Formula y resuelve problemas matemáticos, aplicando diferentes enfoques.
3.
Explica e interpreta los resultados obtenidos mediante procedimientos matemáticos y los contrasta con modelos establecidos o situaciones reales.
4. Argumenta la solución obtenida de un problema, con métodos numéricos, grácos, analíticos analíti cos o variacionales, v ariacionales, mediante el lenguaje verbal, verbal, matemático matemático y el uso de las tecnologías de la información y la comunicación.
5. Analiza las relaciones entre entre dos o más más variables de de un proceso social o natural para determinar o estimar su comportamiento.
6. Cuantica, representa y contrasta experimental o matemáticamente las magnitu des del espacio y las propiedades físicas de los objetos que lo rodean. 7.
Elige un enfoque determinista o uno aleatorio para el estudio de un proceso o fenómeno, y argumenta su pertinencia.
8. Interpreta tablas, tablas, grácas, grácas, mapas, mapas, diagramas diagramas y textos con símbolos matemáticos y cientícos. ESTRATEGIA DIDÁCTICA Para contribuir al desarrollo de las sesiones de aprendizaje en el aula, se estableció una estrategia que permita integrar los elementos del programa de la asignatura,
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Se le denomina estrategia en el sentido de su exibilidad, ya que no pretende ser un algoritmo que el docente deba seguir al pie de la letra, sino que puede adaptarlo a las características propias del c ontexto en el que se desarrollan las sesiones de aprendizaje. La estrategia consta de siete pasos o etapas que deberán conocerse en las primeras sesiones, para un mejor desarrollo de los bloques. Los pasos se listan y describen a continuación:
•
Dinamización.
•
Contextualización.
•
Problematización.
•
Formación, adquisición, desarrollo y construcción de competencias.
•
Síntesis.
•
Realimentación.
•
Evaluación de la competencia.
Dinamización En el proceso de construcción del aprendizaje, es indispensable para el fa cilitador tener evidencia de los aprendizajes previos que el alumno ha adquirido y considerar que es a partir de éstos que se desarrollarán los nuevos, motivando a la colaboración del estudiante en el mismo proceso.
Contextualización En el desarrollo de competencias es necesario el aprendizaje contextual, es decir, presentar elementos a través de escenarios que le sean signicativos a los estudiantes. La contextualización deberá realizarse al inicio de cada bloque en los que se organizan los contenidos en los programas de estudio.
Problematización En el modelo de competencias que la RIEMS establece, el contenido toma un signi-
cado primordial al acercarnos a él, a través, de su aplicación en la vida cotidiana, por tanto, la problematización debe estar presente a lo largo de toda la estrategia en el aula.
Formación, Forma ción, adquisición, desarrollo y construcción de competencias Etapa en la cual el facilitador, a partir de diversas experiencias de aprendizaje, muestra el quehacer del estudiante para lograr las competencias. En esta etapa de la estrategia, estudiantes y docentes deben estar pendientes del proceso de asimilación. Galperin lo describe como un proceso de etapas y no como un fenómeno inmediato. Las distintas etapas del proceso de asimilación que el alumno experimenta para desarrollar el aprendizaje son: la etapa de motivación, la cual debe fomentarse y mantenerse durante todo el curso, recordemos que si un alumno no está motivado, difícilmente aprenderá. La segunda etapa de este proceso es la
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La BOA puede llevarse a cabo de varias formas, cubriendo tres aspectos importantes: la orientación al alumno, que como ya dijimos debe estar precedida por una buena carga de motivación, dicha orientación puede ser de dos tipos, completa en la que el maestro le proporciona al alumno todos los aspectos de un contenido, e incompleta en la cual se dejan ciertos aspectos de un contenido para que el alumno pueda descubrir o investigar por sí mismo. La generalidad es otro aspecto importante en la constitución del BOA, que puede ser concreta o gene ralizada, es decir, el docente puede mostrar hechos concretos relativos a algún contenido, o puede abarcar el mismo contenido por medio de hechos generales, que tengan alguna relación con el concepto que se expone al alumno.
El modo de obtención es el último de los aspectos que incluye la BOA. Éste se presenta de dos formas: pre-elaborada pre-elaborada e independiente. En la primera, el alumno llega a obtener el aprendizaje de manera conjunta con el facilitador y en la segunda, los alumnos adquieren el conocimiento en forma independiente. Síntesis
Actividad que permite integrar los aprendizajes del estudiante a través de eviden cias de conocimiento, desempeño, producto y actitud de manera que el docente cuente con estrategias para la evaluación formativa logrando involucrar al estudiante en procesos de coevaluación. Evaluación de la competencia Para llevar a cabo la evaluación sumativa de las competencias que se indican en los programas de estudio, se contempla esta etapa la cual debe verse como parte del proceso, es decir, no debe en ningún momento separarse de la formativa. La mejor forma de lograr esta unidad será integrando un portafolio de evidencias de aprendizaje.
Dinamización
Síntesis
Contextualización
Realimentación
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Contenido Bloque I: Aplicas la diferencial en estimación de errores y aproximaciones de variables en las ciencias exactas, sociales, naturales y administrativas 3 Sesión 1: La diferencial y sus aplicaciones Diferenciales Aproximaciones de radicales por variables Aproximación de raíces por el método de Newton Estimación de errores
5 6 8 11 14
Bloque II: Determinas la primitiva de una función e integras funciones algebraicas y trascendentes como una herramienta a utilizar en las ciencias exactas, sociales, naturales y administrativas 19 Sesión 1: Primitivas. La integral indefinida. Fórmulas de integración inmediata Integral indefinida Fórmulas de integración directa
20 22 24
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Bloque III: Calculas e interpretas el área bajo la curva en el contexto de las ciencias exactas, sociales, naturales y administrativas 65 Sesión 1: Sumas de Riemann y la integral definida Sumas de Riemann y el problema del área La integral definida como sumas de Riemann
Sesión 2: El teorema fundamental del cálculo Integración aproximada Teorema fundamental del cálculo
66 68 77
82 83 85
Bloque IV: Resuelves problemas de
aplicación de la integral denida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas Sesión 1: Áreas y volúmenes generados por curvas
99 100
Áreas por integración y áreas entre curvas Volúmenes de sólidos de revolución
103 111
Sesión 2: Otras aplicaciones de la integral
120
Aplicaciones a la Economía y Biología
134
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Bloque I: Aplicas Aplicas la diferencial en estimación de errores y aproximaciones de variables
en las ciencias exactas, sociales, naturales y administrativas Objetos de aprendizaje •
La diferencial.
•
Aproximaciones Aproximacio nes de variables.
•
Estimación de errores.
Desempeños del estudiante •
Calcula e interpreta interpreta aproxima aproximaciones ciones de la derivada derivada de modelos modelos matemáticos matemáticos relativos a diversas diversas disciplinas, a partir de su representación gráca y la determinación de su diferencial.
•
Aplica la diferencial para determinar el error presente en el resultado de la medición de una magnitud en diferentes situaciones.
Competencias a desarrollar •
Interpreta grácamente el modelo modelo matemático matemático de fenómeno fenómeno de su entorno entorno y aproxima el comcomportamiento de su derivada a partir del cálculo de la diferencial.
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Bloque I Dinamización En la parte nal de tu travesía en el bachillerato has tenido la oportunidad de com com-prender y desarrollar las diferentes habilidades y destrezas propias de tu perl. Más aún, en esta especialidad te has adentrado al manejo de herramientas matemáticas y analíticas que te permitirán relacionarte con las carreras anes a la especialidad. Sobre este tenor quizás ya estés orientado al estudio superior que te dirigirás, razón por la cual es recomendable que medites sobre los alcances que has logrado en estos 5 semestres para entrelazarlos con los que se te proporcionarán en este último. Como quedó establecido en la obra del pasado semestre, el estudio del cálculo se puede dividir en dos grandes ramas:
Cálculo
Diferencial
Integral
En este semestre nos enfocaremos en el estudio del cálculo integral con su
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Aplicas la diferencial en estimación de errores y aproximaciones de variables en las ciencias exactas, sociales, naturales y administrativas
Sesión 1: La diferencial y sus aplicaciones Criterios:
•
Reconozco la denición de diferencial, así como de los elementos que la constituyen según se me presenten.
•
Relaciono los elementos que componen la diferencial, ya sea de forma analítica o gráca.
•
Realizo aproximaciones de incrementos a diversas situaciones y errores pequeños. Promuevo el estudio y aplicación de las diferenciales para resolver diferentes situaciones hipotéticas o reales.
• •
Empleo el diálogo y diferentes puntos de vista para reforzar la solución a problemáticas surgidas.
Contextualización Recuerda que el cálculo provino de grandes problemáticas analizadas analizadas por diferentes estudiosos de la antigüedad y de tiempos modernos, entre tales retos guraban los
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Bloque I Problematización Si nos presentan determinar las raíces de las siguientes ecuaciones: a)
x 2
−
2 x + 1
=
0
b) x + 2 x − 5 = 0 O si nos piden determinar las raíces: 3
c)
3
27
d)
4
9
Sin el uso de calculadora quizás nos veríamos en predicamentos, pero al analizarlas con detenimiento quizás notemos que los incisos a) y c) son directos, dándonos las raíces de la ecuación x = 1. De forma instantánea la raíz cúbica de 27 es igual a 3. ¡Sin calculadora! Pero qué sucede con las raíces de los incisos b) y d). En primer lugar sin calculadora se nos complica determinar estos valores, pues aparentemente nos darán valores decimales. A n de obtener estos e stos tipos de raíces, o más bien aproxima aproximaciones ciones de ellas sin calculadora, en la antigüedad se utilizaban métodos numéricos recurriendo a la derivada. Considerando estas bases y nociones comenzaremos nuestro recorrido con el contenido aplicativo del cálculo.
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Aplicas la diferencial en estimación de errores y aproximaciones de variables en las ciencias exactas, sociales, naturales y administrativas Por otro lado tenemos la denición de la diferencial de la variable dependiente: dependiente: Defnición : Si una función f está denida por la ecuación y entonces la diferencial de x de x , será:
=
f ( x ) ,
dx = ∆x
En donde esta se denota por dx . Además x está está en el dominio de f´ y y ∆ x es un pequeño incremento arbitrario de x . Con estas deniciones podemos dar a conocer la relación entre las diferenciales de las variables respectivas: dy = f ´( x )dx
Esto señala que la diferencial de una función es igual al producto de su derivada por la diferencial de la variable independiente .
Se nota también que dy depende de los valores de x y y de dx . Si se tiene el valor de dx y x es es tomado del dominio de la función, entonces dy podrá ser calculado. Consideremos para una mejor aclaración, una interpretación geométrica de lo que estamos deniendo.
Es necesario que repases las defniciones de incrementos ∆ x y ∆ y
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Bloque I dy representa la cantidad que la línea
tangente aumenta o disminuye
conforme x cambia cambia por una cantidad dx . ∆ y representa representa la cantidad que la curva y=f(x) aumenta o disminuye conforme x cambia cambia por una cantidad dx . Consideremos un ejemplo: Ejemplo 1:
Halla el valor de
dy cuando
y
x + 3
=
y si x = 2 y ∆ x = 0 .04
Solución:
Primero determinamos dy
= f ´( x )dx : 1
dy = 2
Ahora como x = 2 y
∆ x = dx = 0 .04
1
dy = 2
x + 3
x + 3
dx =
dx
se tendrá que: 1
2 2+3
0.04
0.04 =
=
0. 008944271
2 5
Las diferenciales no sólo nos sirven para calcular incrementos sino también
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Aplicas la diferencial en estimación de errores y aproximaciones de variables en las ciencias exactas, sociales, naturales y administrativas Para adoptar este proceso es mejor realizar los dos ejemplos previamente citados. Ejemplo 2:
Encuentra mediante aproximación de radicales las siguientes raíces (sin calculadora). a) b)
65 3
340
Solución:
Vamos a consolidar lo descrito anteriormente de manera que obser va en cada inciso los pasos realizados.
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Bloque I Lo cual es una buena aproximación al valor real hecho con calculadora: 6.9795532047… Cabe señalar que los cálculos realizados en estos incisos se determinaron sin calculadora, salvo las operaciones para vericar los resultados de las aproximaaproxima ciones. Con el n de visualizar estos aspectos presentamos la gráca del inciso a.
f(65)=√65=? dy
y
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Aplicas la diferencial en estimación de errores y aproximaciones de variables en las ciencias exactas, sociales, naturales y administrativas
Aproximación de raíces Aproximación por el método de Newton
Como te comentaba al inicio de la sesión, las diferenciales tienen diversas aplicaciones, entre ellas obtener las raíces de funciones. A este respecto Sir Isaac Newton ideó un algoritmo que hasta ahora lo utilizan en sus programas las calculadoras cientícas. El método se describe a continuación:
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Bloque I Solución:
Las representaciones grácas de cada una de estas ecuaciones son: y 2.5
x 3+2x-5 x 2-2x+1
2
1.5
1
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Aplicas la diferencial en estimación de errores y aproximaciones de variables en las ciencias exactas, sociales, naturales y administrativas b) En esta función y según la gura 1.3, tomaremos el valor inicial de x = 1.3 que es el que aparentemente nos da la gráca. Usaremos cuatro decimales para ser homogéneos en las tres iteraciones a realizar. 1
x 2
=
x3
=
x 4
=
x 1
−
x 2
−
x 3
−
g( x 1 ) g´( x 1 ) g( x 2 ) g´( x 2 ) g( x 3 )
3
=
=
1.3 −
[(1.3)
+
2(1.3) − 5 ] 2
[3(1.3)
1.3287 −
+
2]
[(1.3287 )3
2
[(1.3282 )
+
1.3287 ......
2(1 .3287 ) − 5 ]
[3(1.3287 ) 3
= 1.3282 −
+
=
+
2]
2(1 .3282 ) − 5 ] 2
=
1.3282 ...
=
1.3282...
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Bloque I
Estimación de errores
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Aplicas la diferencial en estimación de errores y aproximaciones de variables en las ciencias exactas, sociales, naturales y administrativas 1.
Determinen el peso aproximado de un tubo de metal de 6 m de largo si su
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Bloque I Realimentación
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Aplicas la diferencial en estimación de errores y aproximaciones de variables en las ciencias exactas, sociales, naturales y administrativas
Rúbrica del bloque
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Bloque II Los ejemplos indicarán los pasos pertinentes en el uso de estas claves de integración de funciones trigonométricas con potencias. Ejemplo 11:
En cada inciso, integrar las potencias de funciones trigonométricas:
∫ cos 4 xdx sen n xdx ∫ se ∫ sen x cos d ∫ sen 3 x cos 3 xdx
a)
2
5
b) c)
2
d)
3
3
5
e) ∫ tan 2 x sec 2 xdx sen n3 x ⋅ se sen n2 xdx f) ∫ se 3
4
Solución:
Para cada inciso recalcaremos las identidades utilizadas, así como los casos que resulten representados. a) En este caso no se puede integrar de forma directa, pero si realizamos la transformación por identidades trigonométricas, el panorama será más factible. Utilizaremos Utilizaremos el hecho dado por la quinta identidad que recalqué al inicio de esta sección, a saber, que cos2 x = 12 (1 + cos 2 x ). Analizándola, esta se adaptaría del siguiente modo (tomando en cuenta que el argumento es 4 x ):): cos2 4 x = 12 (1 + cos 2( 4 x )))) = 12 (1 + cos 8x ).
Ten presente los argumentos de las identidades para no causar conictos al momento de su empleo en las integrales.
De esta manera nuestro inciso quedará justicado como:
∫ cos
2
4 xdx =
∫
1 2
(1 + cos 8 x )dx =
∫ dx + ∫ cos 8 xdx
1 2
1 2
Llevando a cabo el ajuste en la diferencial dx de de la segunda integral se obtendrá que:
∫ cos
2
4 xdx
=
∫ dx + ∫ cos 8xdx = ∫ dx + ∫ cos 8x 8dx
1
1
1
1 1
2
2
2
2 8
Así que hemos llegado a integrales que se pueden resolver con los métodos directos ya antes señalados en el bloque anterior.
∫ cos
2
4 xdx
=
∫ dx + ∫ cos 8 x 8dx =
1
1 1
1
2
2 8
2
x +
1 16
sen8 x
+ C
b) Como en el caso 2, separamos en potencias cuadradas al factor es decir:
2
sen se n x
,
∫ sen dx = ∫ (sen x ) sen xdx 5
2
2
Ahora realizamos el cambio del factor gonométrica y se logra tener:
∫
5
dx
∫ (
2
2
)
xdx
∫ (1
2
)
2
2
sen se n x
xdx
con la primera identidad tri-
∫ (1
2
2
+
4
)s
xdx
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Determinas Det erminas la primitiva primitiva de una función función e integras integras funciones funciones algebraic algebraicas as y trascendentes trascendentes como una herramienta a utilizar en las ciencias exactas, sociales, naturales y administrativas Esto es: sen n dx = − cos x + ∫ se 5
c)
3
3
cos x
− 15 cos 5 x +
Se presenta a coseno con potencia impar de manera que se v islumbra el caso 1, en donde se reordena la potencia de coseno de la forma siguiente:
sen x cos dx = 2
2
sen x (cos x ) cos xdx =
3
2
2
sen x (1 − sen x ) cos xdx = 2
sen x c os xdx − sen x cos xdx
2
2
4
Integrando por la regla de la cadena para integrales llegamos a la solución: sen n x cos ∫ se 2
dx =
1 3
sen se n x − 3
1 5
5
sen se n x
+ C
d) Este presenta claramente el caso 1, pero con el hecho de poder realizar cualquiera de las conversiones trigonométricas señaladas (sólo una); de manera que al realizar cualquiera de ellas se obtiene una respuesta correcta aunque aparentemente no sean iguales. Realizar Realizaremos emos sólo una de tales opciones que nos aventaja este inciso con el caso 1. Tomaremos la descomposición descomposición de la función seno, de manera que:
∫ sen 3 x cos 3
∫
∫
3 xdx = ( sen 3 x sen )sen3 x cos 3xd x dx = (1 − cos 3 x )sen 3 x cos 3 xdx
5
2
5
2
5
Realizando las operaciones correspondientes separamos las integrales de Realizando manera que se obtienen las formas directas de integración:
∫ sen 3 x cos 3
5
∫
∫
∫
7 5 3 xdx = cos5 3xse sen sen3 x n3 xdx − cos 3 xsen 3 xdx = − 13 cos 3x ( − sen x )3dx +
∫ cos
1 3
7
3 x ( −sen3x )3dx
Integrando esto nos da la solución del problema:
∫ sen 3 x cos 3
5
3 xdx
= − 181 cos6 3x + 241 cos8 3 x + C
Para este caso 3 descomponemos la función secante de manera que surja una potencia cuadrada de ella y luego que realicemos posteriormente la sustitución correspondiente, como sigue:
∫ tan
3
∫
∫
2 x sec 2 xdx = tan 2x sec 2 x (sec 2 x )dx = tan 2x sec 2x (1 + ta t an 2 x )dx 4
3
2
2
3
2
2
Reordenando los componentes de la última expresión e integrando se obtiene:
∫ tan
3
∫
2 x sec 2 xdx = 12 tan 2 x (sec 2 x 2dx ) + 4
3
2
1 2
∫ tan
5
2 x (sec 2x 2dx ) = 2
1 8
1 tan 2 x + 12 tan 2 x + C 4
6
e) Enestecasono recaedirectamenteenloscasosvistos,sinembargo,podemos tomar la identidad trigonométrica sen x cos y = [sen( x − y ) + sen( x + y ) ] ya vista anteriormente. De manera que para el integrando de sen3 x ⋅ sen2 xdx podemos reordenarlo de este modo: 1 2
∫ sen xdx + ∫ se x sen n3 x ⋅ se sen n2 xdx = ∫ [se n(3 x − 2 x ) + sen( 3x + 2 x )]dx = sen n5 xdx ∫ se 1
1
1
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Bloque II Hasta aquí hemos observado sólo unos pequeños ejemplos de la aplicación de las técnicas de integración de potencias de funciones trigonométricas. Recalco lo que señalé al inicio de esta parte, que la práctica podrá darte los mejores elementos elem entos para hacerte diestro en el empleo de este método de integración cuando la situación lo requiera.
Actividad 2 Realiza la integración del inciso c de ejemplo anterior al realizar la sustitución 1 − cos 3 x que se procede después de descomponer el factor coseno. Compara tu resultado con el obtenido en el ejemplo al descomponer el factor seno. En caso de que diera tu respuesta con la previamente obtenida, ¿es signo de error en tu proceso de integración?, ¿por qué? Justica tus respuestas de manera que puedas persuadir a tu docente sobre tu proceso de integración que realizast realizaste. e. 2
Integración por fracciones parciales Recuerda que una fracción propia es aquella en la que el grado del polinomio del numerador es menor al grado del polinomio del denominador. Por el contrario una fracción impropia puede verse con la suma de un polinomio y una fracción propia, por ejemplo x 3 = x 2 +1
x − x
x
2
+1
.
En tu curso de Temas Selectos de Matemáticas 1 te han provisto de los recursos necesarios para descomponer una fracción propia en sus fracciones parciales. Es aquí donde se observa la transversalidad de las competencias genéricas en matemáticas, ya que estas se desarrollan en Temas Selectos de Matemáticas 1 y en Cálculo Integral. Manejaremos en esta sección con fracciones polinomiales propias y simplicadas. Cabe recalcar que todo polinomio con coecientes reales puede ser exex presado como un producto de factores reales lineales de la forma (ax + b) y factores cuadráticos irreducibles de la forma (ax 2 + bx + c ) , dando a entender que un elemento cuadrático de la forma anterior es irreducible en el campo de los números reales si se cumple que b − 4ac < 0 . Se observan entonces cuatro casos pertinentes según los factores del denominador. Por esto se tienen los siguientes casos: 2
Caso I. Factores lineales distintos
Por cada factor lineal (ax + b) que aparezca una sola vez en el denominador de la función racional propia le corresponde una sola fracción simple de A la forma , en donde A es una constante por hallar. ax + b
Caso II. Factores lineales repetidos
Por cada factor lineal (ax + b) que aparezca n veces en el denominador de la función racional propia le corresponde una suma de n fracciones simples A1
de la forma ax + b tantes por hallar.
+
A2 2
( ax + b)
++
An
( ax + b)
n
, en donde A , A , …, y A son cons1
2
n
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Determinas Det erminas la primitiva primitiva de una función función e integras integras funciones funciones algebraic algebraicas as y trascendentes trascendentes como una herramienta a utilizar en las ciencias exactas, sociales, naturales y administrativas Caso III. Factores cuadráticos distintos
Por cada factor cuadrático irreducible en los reales ax + bx + c que aparezca una sola vez en el denominador de la función racional propia le co2
rresponde una sola fracción simple de la forma constantes a determinar.
Ax + B 2
, donde
ax + bx + c
A y B
son
Caso IV. Factores cuadráticos repetidos
Por cada factor cuadrático irreducible en los reales ax + bx + c que aparezca n veces en el denominador de la función racional propia le corresponde una suma de fracciones simples de la forma 2
A1 x + B1
+
2
ax + bx + c
…,y
Bn
A2 x + B2 2
2
(ax + bx + c )
++
An x + Bn n
2 ( ax + bx + c )
, donde
A1 , A2 ,
…,
An
, B1 , B2 ,
son constantes a determinar.
Pudieran preguntarse, ¿qué relación tiene el uso de las fracciones parciales con la integración? Esta relación se presenta porque la fracción polinómica original se descompone en fracciones simples que pueden ser integradas por los métodos convencionales ya vistos. Siguiendo estas líneas es tiempo de remarcar un ejemplo que explore cada uno de los casos previstos. Cabe señalar desde ahora que me evitaré el desarrollo engorroso del álgebra. Ejemplo 12:
En cada inciso determina la integral solicitada usando las fracciones parciales. a)
∫ s t
b) ∫
s +1
+ s2 − 6 s
4
− t 3 − t − 1 dt 3 2 t − t
x
3
x
5
c) ∫ x d)
∫
ds
3
+ x 2 + x + 2 dx 4 + 3x 2 + 2 − x 4 + 4 x 3 − 4 x 2 + 8 x − 4 dx 2 3 ( x + 2)
Solución:
En cada uno de los incisos nos evocaremos al benecio de la descompodescomposición en fracciones parciales y su relación con la integración, de manera que no nos detendremos tanto en el paisaje algebraico. a) El denominador factorizado quedará de la siguiente manera: s3
+
s2
−6
s = s( s + 3)( s − 2)
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Bloque II Ordenando los factores de esta igualdad obtendremos una serie de ecuaciones como sigue: s +1
As( s + 3)( s − 2)
=
s s
+
1
s +1
=
=
A( s
Bs( s + 3)( s − 2)
+
s
+
3
+
Cs( s + 3)( s − 2) s−2
3)( s − 2) + Bs( s − 2) + Cs( s + 3)
+
( A + B + C )s2
+
( A − 2B + 3C )s
( 6 A)
+ −
A + B + C = 0 A − 2B + 3C = 1 −6 A = 1
Resolviendo el sistema presentado se obtienen los valores de las constantes A ,B e C = y así las fracciones quedarán es este orden: =
−
1
6
=
−
2
3
15
10
s +1 s 3 + s2
1
−
6s
−
=
6
s
− +
2
3
15
s+3
+
10
s−2
La integración consecuente es:
∫ s
s +1 3
+ s − 6s 2
ds
ds
= − 16 ∫
s
− 152 ∫
ds
+ 103 ∫
s+3
ds
s−2
= − 16 ln s − 152 ln s + 3 + 103 ln ln s s − 2 + C
b) Se presenta una fracción impropia que al dividir nos arroja el resultado siguiente: t4
−
t
t 3 − t − 1 3
−
2
t
=
t −
t + 1 t 3 − t 2
t + 1
La porción a trabajar e fracciones parciales consta de t − t con factorización del denominador t 3 t 2 t 2 (t 1) que nos recae en el caso 2. 3
−
t + 1 t
3
−
2
t
Que arroja la igualdad sistema:
=
t
=
t + 1 t
2
+
(t − 1)
2
−
A =
t
B +
2
t
C +
t − 1
1 = At( t − 1) + B( t − 1) + Ct2 y
de aquí se forma el
A + C = 0 − A + B = 1 −B = 1
Dándonos las soluciones A 2 , B 1 y C = in = 2 . Conrmando que el integrando se puede separar en fracciones parciales: =
−
=
−
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Determinas Det erminas la primitiva primitiva de una función función e integras integras funciones funciones algebraic algebraicas as y trascendentes trascendentes como una herramienta a utilizar en las ciencias exactas, sociales, naturales y administrativas Resolviendo cada integral con los métodos ya vistos se concluye que:
∫ c)
t
4
− t 3 − t − 1 dt dt dt dt = ∫ tdt + 2 ∫ + ∫ 2 − 2∫ = 3 2 t t − 1 t − t t
1 2
t
2
+ 2 ln t − t − 2 ln t t − 1 + C
Factorizando el denominador se tiene que x + 3x 2 + 2 = ( x 2 + 1)( x 2 + 2) , notando que se trata claramente del caso 3. Entonces: x
3
x
4
2
x
+
+
+ 2
3 x
x + 2 +
2
Ax =
2
x
+
B
+1
Cx +
+
D
+
2
2
x
Agrupando esta igualdad se deduce el sistema: A + C = 1 B+D =1 2 A + C = 1 2B + D = 2
Cuyo resultado es A = 0 , B = 1 , forma en:
1 + x 2 + x + 2 x dx = ∫ dx + ∫ dx 4 2 2 2 x + 3 x + 2 x + 1 x + 2
x
∫
C = = 1 y D = 0 . Con esto la integral se trans-
3
Tomando los medios de integración vistos concluimos lo que se presenta a continuación: 1 2 x + x 2 + x + 2 dx = ∫ dx + 12 ∫ dx = arctan x + 12 ln 4 2 2 2 x + 3x + 2 x + 1 x + 2
x
∫
3
x
+ 2 + C
2
d) Para este inciso del caso 4 tenemos: x
5
−
x
4
+
4x3 2
( x
− +
4x 2
+
3
2)
8 x − 4
Ax =
2
x
+ +
B
2
Cx +
2
( x
+ +
D 2
2)
Ex +
2
( x
+ +
F 3
2)
Al realizar las operaciones pertinentes llegaremos a un sistema de ecuaciones como el que se muestra:
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Bloque II Ordenándolas apropiadamente apropiadamente podremos integrar cada una de estas fracciones parciales: ∫
x
5
− x 4 + 4 x 3 − 4 x 2 + 8 x − 4 dx = ( x 2 + 2)3
∫ x
1 2
2 x
2
+2
dx −
∫ x
2
dx
+ ( 2 )2
+ 2 ∫ ( x 2 + 2)−3 2 xdx = 12 ln x 2 + 2 −
1 2
arctan
x
2
− ( x 2 + 2)−2 + C
Examinando estos ejemplos comprendemos que el poder de las fracciones parciales le otorga al cálculo integral en la integración de fracciones polinomiales propias. Tras este análisis es conveniente entrar en el desarrollo de la siguiente actividad.
Actividad 3 En binas resuelvan las dos integrales de manera que identiquen a cuál de los 4 casos vistos pertenece cada una de ellas.
∫ θ
a)
∫
b)
3θ + 5 3
α
− θ 2 − θ + 1
+3 d α + 2α 2 + 1
2α 4
d θ
2
Integración por sustitución trigonométrica Nuestro método nal de integración se reere a integrandos que poseen cualquiecualquiera de las siguientes expresiones con radicales cuadráticos: a − b x , a + b x o b x − a , donde a y b son constantes arbitrarias. arbitrarias. 2
2
2
2
2
2
2
2
2
Para ello se emplea una sustitución apropiada que posee elementos trigonométricos mediante un cambio de variable, de manera que se integra la expresión en estos términos trigonométricos para luego regresar la expresión integrada a su variable original. Se realiza este cambio de variable a expresiones trigonométricas con el n de obtener una expresión más apropiada o sencilla para integrar con los métodos trigonométricos vistos.
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Determinas Det erminas la primitiva primitiva de una función función e integras integras funciones funciones algebraic algebraicas as y trascendentes trascendentes como una herramienta a utilizar en las ciencias exactas, sociales, naturales y administrativas Caso
Sustituir
Para obtener
Puesto que a 2+b 2 x 2
a
x
+ b2 x 2
2
=
a b
tan z
bx
a sec z
z a
bx 2
b x
2
x
− a2
=
a b
sec
2 2
z
2
b x −a
a tan z
z a
Esta tabla se comprende que al sustituir por ejemplo la expresión x = en a
a 2
−
− b2 x 2
2
2
b x
2
=
a b
senz
se obtendrá: a
2
−
2
b ( ab sen z )
2
=
a
2
−
2
2
a sen z
=
2
a (1
−
cos z
=
sen se n 2z)
a
2
2
−
2
2
a cos z
=
=
a cos z
2
b x
donde Es decir a b x a cos z de como se observa a en el triángulo rectángulo. A partir de este último se pueden determinar las demás funciones trigonométricas para el ángulo z . 2
2
−
2
=
El método consiste como se indicó de forma implícita, en: • •
hacer el cambio de variable obteniendo una relación trigonométrica en términos de z. integrar la relación trigonométrica con los métodos vistos.
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Bloque II Solución:
Abarcaremos cada inciso que corresponde a cada uno de los 3 casos descritos en la tabla. a) Aparentemente no se posee una expresión de la forma a − b x . Pero tras un ajuste de las potencias convertidas a expresiones radicales se tiene: 2
∫ ( 4 − x
3 2 −2
) dx =
dx
∫ ( 4 − x ) 2
∫ (
=
3 2
2
2
dx
4 − x )
2 3
En donde, para este caso las variables a = 4 y b = 1 , o sea a = 2 y b = 1 . Ya que estamos en el primer caso, es necesario tomar la sustitución sen z es decir: x 2
=
2
a
b
x
=
La tabla indica que el radical
2 sen z
4
− x 2
nos dará
a cos z
, o sea,
2cos z
.
Con esto nos jamos en la integral que necesitamos sustituir los siguientes elementos: dx y ( 4 − x 2 )3 de los cuales sólo nos resta calcular dx . Esto se halla al derivar x 2 sen z , de donde: =
dx = 2cos zdz
Sustituyendo estos elementos en la integral la transformamos en:
∫ ( 4 − x ∫ ( 4 − x
3 2 −2
) dx =
dx
∫ ( 4 − x ) 2
3 2 −2
) dx =
∫(
∫ (
=
3 2
dx
=
4 − x )
2 3
dx
4 − x )
2 3
2 cos zdz
∫ (2 cos z )
3
Este integrando ya es de por sí de forma trigonométrica e involucra a la variable z . La reordenamos convenientemente e integramos:
∫ ( 4 − x
3 2 −2
) dx =
∫
2 cos zdz 3
2 cos zdz
= ∫
3
=
1 4
∫
dz 2
= 14 ∫ sec2 zdz = 14 tan z + C
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Determinas Det erminas la primitiva primitiva de una función función e integras integras funciones funciones algebraic algebraicas as y trascendentes trascendentes como una herramienta a utilizar en las ciencias exactas, sociales, naturales y administrativas Que sustituyéndolo en la solución de variable z llegamos llegamos a la solución con la variable original x .
∫ (4 − x
3 2 −2
) dx = 14 tan z + C =
1
x
4
4 − x
x
+ C =
2
b) Claramente se trata del caso 2, pues la expresión Aquí entonces se vislumbra que a = 9 y que b b = 2 . 2
Con esto consideramos x =
a b
tan z
=
+ C
4 4 − x
2
está presente. = 4 , por lo que a = 3 y a
2
+ b2 x 2
2
.
3
tan z
2
Para obtener en vez del radical, la expresión 3 sec z Ya que
dx
= x
3 2
entonces las sustituciones serán:
2
sec zdz
=
3 2
,
sec z dx
=
y
2
3
sec zdz
2
4 x
2
+
9
=
3 sec z
La integral se transforma en:
∫ x
dx
4 x + 9 2
= ∫ 3
3 2
sec2 zdz
sec zdz
( 2 t a n z ) (3 s ec ec z )
= ∫
3 tan z
=
1 cos z
∫
1 3
sen z cos z cos
dz =
cos z
∫ sen z cos z dz = ∫ csc zdz =
1 3
1 3
1 3
ln csc z − cot z + C
Ya determinada la integral sólo resta pasar a la variable original con el uso apropiado de csc z e cot z en el triángulo rectángulo correspondiente lo que nos aporta el siguiente resultado:
∫ x c)
2
4 x
= +9
1
ln csc z 3
− cot z + C =
En este caso se utilizará el cambio Claramente se nota que de por lo que: x
9 + 4 x
2
dx
= 2sec z
dx = 2se zdz 2sec c z ta tan n zd
x
x 2 − 4
=
1
ln 3
a b
sec
2 x
−
3 2 x
+ C Repasa las leyes de los logaritmos para resolver la actividad anterior.
z
logramos obtener que a = 2 y
b = 1 ,
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Bloque II
Actividad 4 En el bloque anterior utilizaste los teoremas 2.4.16, 2.4.18, 2.4.21 y 2.4.22. Utilizando el método de cambio de variable consigue llegar la demostración de 3 de tales teoremas. Una vez concluidas tus demostraciones y cuando tu docente considere pertinente compara tus conclusiones con las de tus demás compañeros para compartir sus experiencias y logren tener las 4 demostraciones.
Síntesis En cada una de los incisos obtén la integral de manera que utilices los métodos vistos en esta sesión, así como los teoremas de integración que analizaste en el bloque anterior. a) ∫ x sec b)
∫ x
c)
∫ s e
2
2
2
3 xdx
=
1 3
x tan 3x
− 19 ln sec 3x + C 3
2 1 − xdx = − 105 (1 − x )2 (15 1 5 x 2 + 12 x + 8 ) + C
−3 s
ds
= − 13 e−3s ( s2 + 23 s + 92 ) + C
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Determinas Det erminas la primitiva primitiva de una función función e integras integras funciones funciones algebraic algebraicas as y trascendentes trascendentes como una herramienta a utilizar en las ciencias exactas, sociales, naturales y administrativas ds
x + 1
n)
∫ s
o)
∫ ( x − 2) dx = ln x − 2 − x − 2 + C
+ 7s + 6
2
=
1 5
ln
+ C
x + 6
2
x
2
4
p) q) r)
x
∫ (1 − x ) dx = − 3
2 x + x + 4 3
∫
∫ e
x
( x + 4 ) dt
− 3e
t
25 − x
=
dx
∫
t) ∫ x
∫
x
1 3e
t
+ 19 ln
a
4
2
2
4 1 − x
t
−3
e
t
=−
a
+ C
2
− x 2 2
a x
x
+ C
− x 2
4
− x 2
x dx
e
x
dx =
dx 2
− 3x − 6 ln 1 − x −
= ln x 2 + 4 + 12 arctan x 2 +
2
s)
2
+ C
2
2
2t
1 2
+ C
x
C
4 2
x
+4
+ C
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Bloque II Mi proyecto del bloque Es momento de trabajar el proyecto del bloque. Te presentamos una situación que te resultará de ayuda para formar y desarrollar las competencias en el presente bloque. También te ayudará con la interacción de las TIC’s. Proyecto:
Vericación Verica ción de teoremas de integración.
Problema:
Cerciorarse de la veracidad de los teoremas de integración vistos en el bloque mediante el uso de la diferenciación.
Duración:
Tres días.
Puntuación:
Competencias:
15 puntos. Argumenta la solución obtenida de un problema con métodos numéricos, grácos, analíticos variacionales, mediante el lenguaje verbal, matemático y el uso de las Tecnologías Tecnologías de la Información y la Comunicación. Para relacionar la operación de integración con lo visto previamente el semestre anterior, a saber: la diferenciación, vas a llevar a cabo las demostraciones de los teoremas relativos a integración, que se señalan en este bloque. Para ello organícense en equipos que tu docente determine, si consideran pertinente deléguense responsabilidades para
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Determinas Det erminas la primitiva primitiva de una función función e integras integras funciones funciones algebraic algebraicas as y trascendentes trascendentes como una herramienta a utilizar en las ciencias exactas, sociales, naturales y administrativas
Realimentación Finalmente considera las siguientes actividades actividades que te servirán de trampolín nal en el desarrollo de las competencias correspondientes. I.
Demuestra los teoremas de integración siguientes: Teoremas 2.4.20, 2.4.21 y 2.4.24.
II.
Utiliza los teoremas de integración pertinentes con el n de obtener las soluciosolucio nes señaladas en cada inciso. a)
∫ xdx =
b)
∫
c)
∫ ( x
d)
∫ x − 1 = ln x − 1 + C
e)
∫ e
f)
3
3 4
4 / 3
x
2
3 x − 1dx = xdx 2
+ 4)
3
=
9
+ C (3x − 1)
3 / 2
+ C
−1 + C 2 4( x + 4 ) 2
dx
∫
4 x
dx
e 2 x
=
1 4
e
2 x
dx 2
4 x
=
+C
1 2
ln e
2 x
+2 +C
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Bloque II 2
c)
∫ x (
d)
∫ a + bt
e)
∫ 1 − cos x dx
f)
∫ (e
g)
∫ x + sen x dx
h)
dx ∫ 16 − se sen n x
i)
∫ 4 y
j)
∫
a
t
2
−
x ) dx
dt
sen x
2 z
+ a2 z )dz
1 + cos x
cos x
2
dy
2
+ 25
dx 2
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Determinas Det erminas la primitiva primitiva de una función función e integras integras funciones funciones algebraic algebraicas as y trascendentes trascendentes como una herramienta a utilizar en las ciencias exactas, sociales, naturales y administrativas d)
∫ sen xdx
e)
∫ e
f)
∫ (e
g)
∫ tan xdx
h)
∫ tan θ sec
4
2 t
cos3tdt
θ
+ 2θ ) 2d θ 3
3 2
4
θ d θ
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Bloque II Evaluación de la competencia Responde la rúbrica correspondiente al proyecto para que analices junto con tu docente y prevengan lo que has de cumplir a lo largo de éste. Rúbrica para la evaluación del proyecto Producto, logro o desempeño
Nivel de logro o desempeño 5
4
3
2
1
Estratégico
Autónomo
Básico
Inicial
Pre-formal
Reconozco la
Reconozco
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Determinas Det erminas la primitiva primitiva de una función función e integras integras funciones funciones algebraic algebraicas as y trascendentes trascendentes como una herramienta a utilizar en las ciencias exactas, sociales, naturales y administrativas
Rúbrica del bloque Te proporciono la rúbrica de este segundo bloque con e l n de que no olvides cuáles son los objetivos a considerar durante el mismo. Te proporciono la rúbrica de este segundo bloque con el fn de que no olvides cuáles son los objetivos a considerar durante el mismo. Rúbrica para la evaluación del proyecto Producto, logro o desempeño
Nivel de logro o desempeño 5
4
3
2
1
Estratégico
Autónomo
Básico
Inicial
Pre-formal
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Bloque II Rúbrica para la evaluación del proyecto Producto, logro o desempeño
Nivel de logro o desempeño 5
4
3
2
1
Estratégico
Autónomo
Básico
Inicial
Pre-formal
Utilizo las diferentes fórmulas de integración inmediatas con el n de obtener las antiderivadas
Con ayuda utilizo las diferentes fórmulas de integración inmediatas con el n de obtener las
Utilizo las diferentes fórmulas de integración inmediatas con el n de obtener
Con ayuda utilizo algunas de las diferentes fórmulas de integración
Con ayuda utilizo pocas o ninguna de las diferentes fórmulas de integración inmediatas con
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Determinas Det erminas la primitiva primitiva de una función función e integras integras funciones funciones algebraic algebraicas as y trascendentes trascendentes como una herramienta a utilizar en las ciencias exactas, sociales, naturales y administrativas Rúbrica para la evaluación del proyecto Producto, logro o desempeño
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4
3
2
1
Estratégico
Autónomo
Básico
Inicial
Pre-formal
Considero las actitudes y razonamientos de mis compañeros de manera
Considero las actitudes o razonamientos de mis compañeros de manera reexiva para llegar a un
Considero las actitudes y razonamientos de mis
No considero las actitudes ni razonamientos de mis compañeros de
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Bloque III Dinamización
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Calculas e interpretas el área bajo la curva en el contexto de las cienci iencias as exactas, social sociales, es, naturales y administrativas
Problematización
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Bloque III Pueden usar los medios necesarios y materiales como reglas, calculadoras,
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Bloque III Evaluación Evaluac ión de la competencia Aquí está la rúbrica del proyecto correspondiente correspondiente a este bloque. Rúbrica del proyecto Producto, logro o desempeño
Conocimientos
Nivel de logro o desempeño 5
4
3
2
1
Estratégico
Autónomo
Básico
Inicial
Pre-formal
Conozco todos los elementos analíticos y grácos que componen una integral denida de acuerdo al contexto en que se me proporciona. Describo el funcionamiento completo del software así como los elementos que lo componen y reconozco otras aplicaciones opcionales.
Conozco algunos de los elementos analíticos y grácos que componen una integral denida de acuerdo al contexto en que se me proporciona. Describo el funcionamiento básico del software así como los elementos que lo componen y reconozco otras aplicaciones opcionales.
Conozco solo los elementos grácos que componen una integral denida de acuerdo al contexto en que se me proporciona. Describo el funcionamiento básico del software así como los elementos que lo componen.
Conozco solo los elementos analíticos que componen una integral denida de acuerdo al contexto en que se me proporciona. Describo el funcionamiento muy escueto del software así como los elementos que lo componen.
No reconozco los elementos analíticos ni grácos que componen una integral denida. Describo solo el funcionamiento escueto del software pero no identico ningún elementodel mismo como los menús.
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Calculas e interpretas el área bajo la curva en el contexto de las cienci iencias as exactas, social sociales, es, naturales y administrativas
Rúbrica del proyecto Producto, logro o desempeño
Habilidades
Nivel de logro o desempeño 5
4
3
2
1
Estratégico
Autónomo
Básico
Inicial
Pre-formal
Represento de forma óptima las integrales denidas solicitadas a la vez de que puedo realizarles ciertos ajustes opcionales para su mejor visualización. Puedo imprimir en presencia de mi docente las integrales denidas que indique y respondo de manera correcta y aporto ideas a las preguntas planteadas al exponer lo realizado.
Represento de forma adecuada las integrales denidas solicitadas a la vez de que puedo realizarles ciertos ajustes opcionales para su mejor visualización. Puedo imprimir en presencia de mi docente las integrales denidas que indique y respondo de manera correcta a las preguntas planteadas al exponer lo realizado.
Represento de forma adecuada las integrales denidas solicitadas y les realizo algunos ajustes con ayuda proporcionada. Puedo imprimir en presencia de mi docente las integrales denidas que indique y respondo de manera correcta a la mayoría de las preguntas planteadas al exponer lo realizado.
Represento de forma elemental las integrales denidas solicitadas sin realizarles ajustes opcionales para su mejor visualización. Con ayuda puedo imprimir en presencia de mi docente las integrales denidas que indique y respondo de manera correcta a algunas de las preguntas planteadas al exponer lo realizado.
No puedo representar de forma adecuada las integrales denidas solicitadas. Con ayuda puedo imprimir en presencia de mi docente las integrales denidas que indique pero no respondo de manera correcta a las preguntas planteadas al exponer lo realizado.
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Bloque III Rúbrica del proyecto Producto, logro o desempeño
Actitudes
Puntaje
Nivel de logro o desempeño 5
4
3
2
1
Estratégico
Autónomo
Básico
Inicial
Pre-formal
Tengo un alto compromiso con mi equipo y con la resolución del proyecto de forma que colaboro con él en todo momento. Mantengo una actitud positiva en todo momento del trabajo además de que expreso mis ideas y aportaciones con un lenguaje digno en todo tiempo. Demuestro un alto interés en el manejo del software dando otras posibles interpretaciones del mismo.
Tengo un alto compromiso con mi equipo y con la resolución del proyecto de forma que colaboro con él en la mayoría del tiempo. Mantengo una actitud neutral en todo momento del trabajo además de que expreso mis ideas y aportaciones con un lenguaje digno en todo tiempo. Demuestro interés en el manejo del software dando otras posibles interpretaciones del mismo.
Tengo un compromiso concreto con mi equipo y con la resolución del proyecto de forma que colaboro con él en la mayoría del tiempo. Mantengo una actitud neutral en todo momento del trabajo además de que expreso mis ideas y aportaciones con un lenguaje básico. Demuestro interés en el manejo del software.
15
12
9
Tengo un bajo compromiso con mi equipo y con la resolución del proyecto de forma que colaboro con él. Mantengo una actitud pasiva en todo momento del trabajo además de que expreso en ocasiones mis ideas y aportaciones. Demuestro poco interés en el manejo del software.
6
Tengo un bajo compromiso con mi equipo y con la resolución del proyectopero no colaboro con él. Mantengo una actitud apática en todo momento del trabajo además de que no expreso ideas niaportaciones. Demuestro apatía y falta de interés en el manejo del software. 3
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Calculas e interpretas el área bajo la curva en el contexto de las cienci iencias as exactas, social sociales, es, naturales y administrativas
Rúbrica del bloque Observa y medita detenidamente cada uno de los niveles de la rúbrica del bloque a n de considerar lo que se debe evidenciar en tu formación en este bloque. Rúbrica para la evaluación del bloque Nivel de logro o desempeño Producto, logro o desempeño
5
4
3
2
1
Estratégico
Autónomo
Básico
Inicial
Pre-formal
Distingo adecuadamente la relación gráca, algebraica y conceptual de integral denida que provenga de una situación hipotética y real. Describo todos los elementos de una suma de Riemann bajo la conexión del área bajo una curva en general. Conocimientos
Comprendo todos los diferentes métodos necesarios para el cálculo de integrales no directas.
Identico claramente cuál o cuáles métodos utilizar para integrar según los datos de las integrales presentadas.
Distingo parcialmente parcialmen te la relación gráca, algebraica y conceptual de integral denida que provenga de una situación hipotética y real. Describo algunos de los elementos de una suma de Riemann bajo la conexión del área bajo una curva en general. Comprendo tres de los diferentes métodos necesarios para el cálculo de integrales no directas. Identico parcialmente cuál o cuáles métodos utilizar para integrar según los datos de las integrales presentadas.
Distingo parcialmente la relación gráca, algebraica y conceptual de integral denida que proviene del ámbito hipotético o real.
Distingo vagamente y con ayuda la relación gráca, algebraica y conceptual de integral denida que provenga de una situación hipotética o real.
Describo algunos de los elementos de una suma de Riemann bajo la conexión del área bajo una curva en particular.
Describo con ayuda algunos de los elementos de una suma de Riemann bajo la conexión del área bajo una curva particular.
Comprendo dos de los diferentes métodos necesarios para el cálculo de integrales no directas.
Comprendo uno los diferentes métodos necesarios para el cálculo de integrales no directas.
Identico parcialmente y con ayuda cuál o cuáles métodos utilizar para integrar según los datos de las integrales
Identico dudosamente y con ayuda cuál o cuáles métodos utilizar para integrar según los datos de las integrales
No distingo la relación gráca, algebraica ni conceptual de integral denida que provenga de situaciones hipotéticas o reales. No describo los elementos de una suma de Riemann bajo la conexión del área bajo una curva. No comprendo ninguno de los diferentes métodos necesarios para el cálculo de integrales no directas. No identico cuál o cuáles métodos utilizar para integrar según los datos de las integrales presentadas.
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Bloque IV: Resuelves problemas
de aplicación de la integral denida en situaciones reales
en el campo de las ciencias exactas, sociales, natur naturales ales y administrativas Objetos de aprendizaje •
Áreas y volúmenes de curvas y de sólidos de revolución.
•
Ley de Newton.
•
Crecimientos exponenciales.
•
Oferta y demanda.
Desempeños del estudiante •
Aplica el concepto concepto de sólido de revolución revolución en el diseño diseño de: envases, depósitos y contenedores contenedores en general, de formas homogéneas y heterogéneas.
•
Aplica las integrales integrales denidas denidas en la solución solución de problemas de leyes de Newton Newton (centro de masa, trabajo realizado realizado por una fuerza, movimiento de partículas) y/ o crecimientos exponenciales, resolviéndolos de manera autónoma utilizando los procesos aprendidos.
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Bloque IV Dinamización Llegamos a la parte nal del camino en donde aplicaremos a problemas del contexto los teoremas fundamentales fundamentales del cálculo así como de la interacción de la integración con diferentes áreas de conocimiento. Es decir no sólo abarcaremos la utilización del proceso de integración al cálculo de áreas de curvas, sino que la aplicación se extenderá al cálculo de volúmenes, longitudes de arco de curvas, centros de masa, etcétera. La parte principal de este bloque radica en la evaluación de las integrales denidas. Es aquí donde estableceremos la conexión de los diferentes saberes recorecorridos en los bloques anteriores y también de tus semestres que has cursado. Recuerda que a lo largo de estos últimos dos semestres deberás de manifestar las competencias disciplinares disciplinares extendidas, así que es oportuno opor tuno recordarte que las consideres durante este último escalafón que estás por dar en el bachillerat bachillerato. o. ¡Éxito en lo que hagas¡
Sesión 1: Áreas y volúmenes generados por curvas Criterios
• •
Comprendo la utilidad de las fórmulas de áreas y volúmenes generadas por una o más curvas. cur vas. Identico las características de las áreas y volúmenes generadas por curvas.
•
Represento de forma gráca el área y volumen generado por una o más curvas, según sea el caso y según sea el eje de simetría.
•
Empleo las fórmulas y métodos para determinar las áreas y volúmenes de rotación de curvas que se me proporcionen.
•
Soy perseverante al momento de emplear las fórmulas de áreas y volú-
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Resuelves problemas de aplicación de la integral definida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas Y
y=f(x)
Área de la región R es: a
ʃ
A= b f(x)dx
X
x=a
x=b
b
∫ f ( x )dx
Figura 4.1 El área de la región se determina con la integral denida A =
a
En esta sección nos abocaremos al inicio de las aplicaciones directas de la integral denida, que surgió con el estudio de los antiguos griegos al querer determideterminar el área área de diferentes guras mediante aproximaciones. En nuestro caso se tratará de área entre grácas de funciones del tipo y = f ( x ) sobre el eje X o o del tipo x = g( y ) sobre el eje Y . Además también tocaremos tocaremos el cálculo de volúmenes que se forma al girar las curvas en torno a diferentes ejes de rotación.
Problematización Para dar inicio consideremos la siguiente situación: Un arquitecto está contemplando desarrollar un proyecto de plaza comercial en donde surge un jardín que tiene una forma muy especial como la siguiente
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Bloque IV
Este arquitecto tiene la responsabilidad de hacerle saber al ingeniero de obra civil del proyecto, el área que contendrá esta jardinera. Algo curioso es que esta jardinera está acotada o delimitada delimitada por dos funciones conocidas. conocidas. Desde el plano plano ararquitectónico se desprende que, vista de la forma anterior, anterior, las funciones son f ( x ) = x para la parte superior y recta; y g( x ) = x 2 para la parte curva inferior. Con estos datos el arquitecto se ve en la disyuntiva de cómo calcular el área de la región que ocupará esta jardinera en la obra. Si en una integral defnida hallamos un área que nos resulta negativa, esto señala
Actividad 1 En parejas reúnete con un compañero para que ambos exploren las posibles ayudas
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Resuelves problemas de aplicación de la integral definida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas
Áreas por integración y áreas entre curvas
Conforme fuimos avanzando en esta obra nos centrábamos cada vez más en que para una función f ( x ) ≥ 0 el área de la región se determinaba por medio de la integral denida con los límites límit es inferior y superior que las rectas del eje X nos nos determinab ban, es decir, ∫ a f (x )d x y el valor de esta integral es positivo. Sin embargo, también se tendrá el caso en que f ( x ) ≤ 0 , por lo tanto se comprende que el valor de la integral b será negativa, o sea ∫ f ( x )dx ≤ 0 así que si deseamos determinar el área de esta funa b ción que se halla debajo del eje X podemos realizar la integral − ∫ a f (x )dx con lo cual nos dará obviamente positiva. Esta a su vez será positiva si se halla a la derecha del eje Y y y negativa si se encuentra al lado izquierdo de este mismo eje. Si la función respectiva cambia de signo en el intervalo descrito entonces el área correspondiente viene dado por la suma de dos o más integrales denidas. Cuando nos topamos con regiones de áreas que resultan estar entre las curvas continuas en [a, b] y = f ( x ) y y = g( x ) con f ( x ) ≥ g( x ) entreb las rectas x = a dx y x = b , el área de estas regiones se se determinan con la integral integral ∫ a [ f ( x ) − g( x )]dx Por el contrario, si se consta de regiones que resulten estar entre las curvas continuas en [c, d ] x = f ( y ) y x = g( y ) con f ( y ) ≥ g( y ) entre las rectas y = c y y = d , d dy . el área de estas regiones se determinan con la integral ∫ c [ f ( y ) − g( y )]dy Toda esta pequeña discusión la resumo en la siguiente tabla: Región R
Gráfco Y
y=f(x)
Fórmula
A su vez es posible que tengamos una función x
= g( y ) al lado de-
recho del eje Y con los límites y = c y y = d con lo que su integral d
será
∫ g( y )dy . c
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Bloque IV Y
y=d
R
Comprendida por x = g( y ) ≥ 0 en [c , d ] , el eje Y , y las rectas y = c y
d
∫ g( y )dy
x=f(y)
c
y = d g( y ) continua en [c , d ]
y=c
X
Y
y=d
Comprendida por x = g( y ) ≤ 0 en [c , d ] , el eje Y , y las rectas y = c y y = d g( y ) continua en [ c , d ]
x=f(y)
d
− ∫ g (y )dy
R
c
y=c X
Y
Comprendida Comprendi da po
f( )
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Resuelves problemas de aplicación de la integral definida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas Para determinar la región de un área con las integrales denidas realizamos: •
•
Una representación gráca donde se visualice la región en cuestión, esto incluyendo el intervalo que de manera explícita o implícita se describa. Este intervalo nos dará los límites inferior o superior para la integral denida. Dibujar un i-ésimo rectángulo de longitud ∆ x que llamaremos fran ja vertical (o ∆ x si el i-ésimo rectángulo tiene su base en el eje Y que llamaremos una franja horizontal). i
i
•
Suponer que el número de rectángulos crece indenidamente para aplicar el segundo teorema fundamental del cálculo de la forma correspondiente de acuerdo a la tabla anterior.
Nota: para determinar el área de una región que por ejemplo cor ta al eje X,
no necesariamente se ha de recurrir a los i-ésimos intervalos con base ∆ x o franjas verticales, sino también en ocasiones, será posible hacerlo con i-ésimos intervalos de longitud ∆ x o franjas horizontales. Esto da a entender que una función de variable d −1 1 y = f ( x ) puede integrarse de la forma ∫ c f ( y )dy , donde x f ( y ) es el despeje de la variable x en y = f ( x ) . Esto se señalará más adelante con un ejemplo. =
−
Por tanto comencemos a realizar algunos cálculos de áreas para compren-
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Bloque IV Estaremos trabajando con franjas verticales. Notamos que la altura del i-ésimo rectángulo representativo puede considerarse como y i así que el rectángulo tendrá un área de ∆i x ⋅ y i (base por altura) , o sea y i ⋅ ∆ i x . De modo que si tomamos el límite siguiente: n
lim ∑ y i ∆ i x ∆ →0
pero como y
=
4x
−
2
x
n
, entonces el área de ese rectángulo será
lim ∑ y i ∆ i x = ∆ →0
i =1
i =1
∫
4
0
4
ydx = ∫ ( 4 x − x 2 )dx 0
En esta última integral denida se toman los límites inferior y superior como 0 y 4 respectivamente, puesto que en ese intervalo [0, 4] se encuentra la región determinada. Algo que te podrá servir como método de ubicación en estos casos será considerar que la altura del rectángulo representativo sea y o sea 4 x − x y la base ∆ x que se “transforma” en dx en la integral denida, por ello esta última tiene la b b forma ∫a (altura)( base) = ∫ a ( y )( dx) . 2
Prosiguiendo con la integral ésta resultará en lo siguiente de acuerdo al segundo teorema fundamental del cálculo:
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Resuelves problemas de aplicación de la integral definida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas La gráca muestra que el en el intervalo [−1, 1] se tiene f ( x ) ≤ 0 y para [1, 3] se contempla que f ( x ) ≥ 0 , por lo tanto, es necesario dividir la región total en dos A será la suma de los áreas regiones R1 y R 2, de donde se comprende que el área total A A1 y A 2 respectivas a las dos regiones descritas. Para la región R1 el valor de la altura del rectángulo representativo representativo es − y i ; mientras que para la región R 2 será de y i . Entonces las integrales correspondientes quedarán de esta manera: 1
1
1
1
−1
−1
−1
−1
A1 = ∫ − ydx = − ∫ ydx = − ∫ (− x 3 + 2x 2 + 5x − 6)dx = ∫ ( x 3 − 2x 2 − 5x + 6)dx 3
3
A2 = ∫ ydx = ∫ ( − x 3 + 2x 2 + 5x − 6)dx 1
1
Utilizando el segundo teorema fundamental del cálculo estas áreas serán: 1
A1
= ∫ ( x 3 − 2 x 2 − 5 x + 6)dx = ( x4 − 23x − 5 2x + 6 x )
A2
= ∫ (− x 3 + 2 x 2 + 5 x − 6)dx = (− x4 +
4
2
−1 3
4
1
Entonces el área total es A = A
1
Ejemplo 3:
3
+
A2
2x 3
=
3
+
16
2
5 x 2
1
−1
37 91 = ( 12 ) − ( −12 ) = 12
3
32 3
− 6 x ) = ( 94 ) − ( −1237 ) = 1
16 3
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Bloque IV Notamos que la región es simétrica respecto al eje X , razón por la cual nos va a ahorrar unos pasos para determinar su área. Considerando a x c omo variable independiente, la gráca se compone de dos partes: y y
=
−
4
−
x
=
4
−
x
para la parte superior del eje X y
para la parte inferior del eje X .
a) Debido a la simetría de esta región no es necesario determinar el área hallada en la parte superior del eje X y y al mismo tiempo la que se encuentra en la parte inferior del mismo eje. Nos basta calcular cualquiera de ella y multiplicar el resultado por dos. En esta ocasión vamos a hallar el área que se encuentra en la parte inferior del eje X , por lo tanto la altura del rectángulo de base ∆ x será será − y i . Notamos también que el intervalo a considerar es [0, 4] en el eje X . i
El área está dada al considerar la función − 1 2
4−
x : 4
A = 2 ∫ −( y )dx = 2∫ −(− 4 − x )dx = 2∫ ( 4 − x ) dx = [− 43 ( 4 − x ) ] = [0] − [− 323 ] = 0 0 0 0 4
4
4
3 2
32 3
b) Para el caso de las franjas horizontales resulta más sencillo, ya que al ser simétrica el intervalo [−2, 2] en el eje Y puede resumirse al intervalo [0, 2]
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Resuelves problemas de aplicación de la integral definida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas Y 5
4
(9,4) 3 0.5 2
1 X
-1
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Bloque IV El área para cada región estará dada por : 3
3 3 A1 = 2 ∫ ydx = 2 ∫ 2 x − 2 dx = [ 23 (2 x − 2 ) ] = 1 1 1 3 2
16 3
9
A2 = ∫ ( y1 − y2 )dx = ∫ [ 2 x − 2 − ( x − 5)]dx = ∫ [ 2 x − 2 − x + 5]dx = [ 13 (2 x − 2) − 12 x 2 + 5x ] = ( 155 ) − ( 796 ) = 6 3 3 3 3 9
9
9
3 2
38 3
d) Para este caso la representación quedará de acuerdo a la gura siguiente: siguiente: 6
La resta en el inte-
∫
grando ( y
y )dx
5
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Resuelves problemas de aplicación de la integral definida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas
Volúmenes de sólidos de revolución Proseguimos con las aplicaciones de la integral denida.
Actividad 3 Imagina que posees un triángulo rectángulo en un cartoncillo con las siguientes medidas:
Eje de revolución
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Bloque IV Representa grácamente los volúmenes que se observarían en ambos casos. Esta actividad nos sugiere que al girar regiones o áreas planas bajo ciertos ejes de rotación podremos generar ciertos volúmenes, por ello para iniciar la parte conceptual de esta sección se dene lo que compone un sólido de revolución. Defnición . Un sólido de revolución es un sólido que se genera al girar una región plana en torno a una recta del mismo plano, esta recta se llama eje de rotación o eje de revolución y puede intersectar o no a la región.
En esta obra emplea-
Por ejemplo, si hacemos girar el triángulo rectángulo anterior de forma vertical en torno al cateto menor, podremos observar un sólido de revolución como el siguiente (vamos a considerar el cateto menor como el eje X positivo positivo y el cateto
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Resuelves problemas de aplicación de la integral definida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas Los sólidos de revolución pueden ser generados en diferentes casos, estos son cuando: •
El eje de revolución forma parte del contorno de la región.
•
El eje de revolución no es parte del contorno de la región.
Para determinar de forma exacta el volumen de un sólido de revolución se recurre a métodos especícos que se describen a continuación. Método de los discos
Este se utiliza cuando el eje de revolución forma parte del contorno de la región y se emplean los siguientes pasos: •
Trazar un diagrama indicando el área de la región, así como un i-ésimo
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Bloque IV El valor de aparece en esta relación, ya que si hacemos girar el rectánπ
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Resuelves problemas de aplicación de la integral definida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas
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Bloque IV Método de las capas
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Resuelves problemas de aplicación de la integral definida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas Solución:
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Bloque IV Podemos observar que será atacado mediante el método de las arandelas
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Bloque IV Rúbrica del bloque Nota los criterios de la rúbrica del bloque de manera que te propongas la meta de subir al menos un nivel de dominio en el cual te halles al inicio del mismo.
Rúbrica para la evaluación del bloque Producto, logro o desempeño
Nivel de logro o desempeño 5
4
3
2
1
Estratégico
Autónomo
Básico
Inicial
Pre-formal
Deno de forma verbal y analítica el teorema fundamental del cálculo con todos los elementos que le corresponden. Interpreto todos los modelos matemáticos presentados en mi contexto relacionados al teorema fundamental del cálculo y a la regla de Simpson. Conocimientos
Comprendo siempre la utilidad de las fórmulas de áreas y volúmenes generadas por una o más curvas e identico las características característic as de las áreas y volúmenes generadas por curvas. Describo diferentes y variadas facetas en las ciencias, en donde sea aplicable la integral denida como medio de resolución de una problemática.
Deno de forma verbal o analítica el teorema fundamental del cálculo con todos los elementos que le corresponden. Interpreto algunos de los modelos matemáticos de mi contexto relacionados al teorema fundamental del cálculo y a la regla de Simpson. Comprendo en ocasiones la utilidad de las fórmulas de áreas y volúmenes generadas por una o más curvas e identico las características característi cas de las áreas y volúmenes generadas por curvas. Describo diferentes facetas en las ciencias, en donde sea aplicable la integral denida como medio de resolución de una problemática.
Deno de forma verbal o analítica el teorema fundamental del cálculo con algunos de los elementos que le corresponden. Interpreto algunos de los modelos matemáticos de mi contexto relacionados al teorema fundamental del cálculo y a la regla de Simpson. No comprendo la utilidad de las fórmulas de áreas y volúmenes generadas por una o más curvas o no identico las características de las áreas y volúmenes generadas por curvas pero no ambas cuestiones. Describo algunas facetas correctas en las ciencias, en donde sea aplicable la integral denida como medio de resolución de una
Deno incorrectamente de forma verbal o analítica el teorema fundamental del cálculo con los elementos que le corresponden. Interpreto con dicultad algunos de los modelos matemáticos de mi contexto relacionados al teorema fundamental del cálculo y a la regla de Simpson. Comprendo parcialmente la utilidad de las fórmulas de áreas o volúmenes generadas por una curva e identico en ocasiones las características característic as de las áreas o volúmenes generadas por curvas. En ocasiones describo facetas parcialmente correctas en las ciencias, en donde sea aplicable la integral denida como medio de resolución de una problemática.
No deno de forma verbal ni analítica el teorema fundamental del cálculo con los elementos que le corresponden. No interpreto algún modelo matemático de mi contexto relacionado al teorema fundamental del cálculo y a la regla de Simpson. No comprendo la utilidad de las fórmulas de áreas y volúmenes generadas por una o más curvas ni tampoco identico las características de las áreas y volúmenes generadas por curvas. No describo faceta alguna en la ciencia, en donde sea aplicable la integral denida como medio de resolución de una problemática.
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Resuelves problemas de aplicación de la integral definida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas Rúbrica para la evaluación del bloque Producto, logro o desempeño
Habilidades
Nivel de logro o desempeño 5
4
3
2
1
Estratégico
Autónomo
Básico
Inicial
Pre-formal
En el principio fundamental del cálculo comparo constantemente la relación entre los elementos que entran en juego para su uso.
En el principio fundamental del cálculo comparo en ocasiones la relación entre los elementos que entran en juego para su uso.
Aplico correctamente la regla de Simpson relacionada al cálculo de un área bajo una curva según lo pedido.
Aplico correctament correctamentee la regla de Simpson relacionada al cálculo de un área bajo una curva en condicione condicioness sencillas.
Represento adecuadamente adecuadam ente y de forma gráca el área y volumen generado por una o más curvas, según sea el caso y según sea el eje de rotación.
Represento adecuadamente y de forma gráca el área y volumen generado por una o más curvas, según sea el caso y según sea el eje de rotación.
Empleo las fórmulas correctamente, correctament e, así como los métodos apropiados para determinar las áreas y volúmenes de revolución de curvas que se me proporcionen. Aplico las integrales denidas requeridas con el n de resolver situaciones relacionadas a las ciencias experimentales o sociales, ya sean hipotéticas o reales.
Empleo las fórmulas correctamente, así como algunos de los métodos para determinar las áreas y volúmenes de revolución de curvas que se me proporcionen. Aplico las integrales denidas requeridas con el n de resolver situaciones relacionadas a las ciencias experimentales o sociales, que sean hipotéticas.
En el principio fundamental del cálculo comparo la relación entre algunos de los elementos que entran en juego para su uso. Aplico con dicultad la regla de Simpson relacionada al cálculo de un área bajo una curva. Represento de forma básica la gráca, el área y volumen generado por una o más curvas, según sea el caso y según sea el eje de rotación. Empleo las fórmulas y métodos de forma inconsistente al momento de determinar las áreas y volúmenes de revolución de curvas que se me proporcionen. Aplico con ciertos errores las integrales denidas requeridas con el n de resolver situaciones relacionadas a las ciencias experimentales o sociales, ya sean hipotéticas o reales.
En el principio fundamental del cálculo comparo en ocasiones la relación entre algunos de los elementos que entran en juego para su uso. Aplico con ciertos errores la regla de Simpson relacionada al cálculo de un área bajo una curva. Represento con ciertos errores la gráca del área y volumen generado por una o más curvas, según sea el caso y según sea el eje de rotación. Empleo las fórmulas y métodos de forma incorrecta al momento de determinar las áreas y volúmenes de revolución de curvas que se me proporcionen. Aplico con errores las integrales denidas requeridas con el n de resolver situaciones relacionadas a las ciencias experimentaless o experimentale sociales, de la forma hipotética.
En el principio fundamental del cálculo no puedo comparar la relación entre los elementos que entran en juego para su uso. No aplico la regla de Simpson relacionada al cálculo de un área bajo una curva. No puedo representar de forma gráca el área y volumen generado por una o más curvas, según sea el caso y según sea el eje de rotación.
No empleo las fórmulas ni los métodos para determinar las áreas y volúmenes de rotación de curvas que se me proporcionen. No aplico las integrales denidas requeridas con el n de resolver situaciones relacionadas a las ciencias experimentales o sociales, ya sean hipotéticas o reales.
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Bloque IV Rúbrica para la evaluación del bloque Producto, logro o desempeño
Nivel de logro o desempeño 5
4
3
2
1
Estratégico
Autónomo
Básico
Inicial
Pre-formal
Utilizo de forma continua el diálogo entre mis compañeros y docente para resolver situaciones presentadas. Mantengo y reejo siempre una actitud positiva frente a las dicultades que se presentan al equipo o de forma individual.
Actitudes
Soy perseverante al momento de emplear las fórmulas de áreas y volúmenes con el n de mantener óptimo ritmo de trabajo. Estoy consciente de mis valores, fortalezas y debilidades con el n de recurrir siempre a la ayuda necesaria de cualquier fuente de manera que conserve el respeto debido a mis compañeros y docente. Reexiono continuamente sobre la aplicación de las integrales en diversas áreas así como de su utilidad en ellas.
Utilizo el diálogo entre mis compañeros y docente para resolver situaciones presentadas. Mantengo siempre una actitud positiva frente a las dicultades que se presentan al equipo o de forma individual. Soy perseverante al momento de emplear las fórmulas de áreas y volúmenes con el n de mantener un buen ritmo de trabajo. Estoy consciente de mis valores, fortalezas y debilidades con el n de recurrir en ocasiones a la ayuda necesaria de cualquier fuente de manera que conserve el respeto debido a mis compañeros y docente. Reexiono ocasionalmente sobre la aplicación de las integrales en diversas áreas así como de su utilidad en ellas.
Utilizo en ocasiones el diálogo entre mis compañeros y docente para resolver situaciones presentadas. Mantengo y reejo ocasionalmente una actitud positiva frente a las dicultades que se presentan al equipo o de forma individual. Soy perseverante al momento de emplear las fórmulas de áreas y volúmenes con el n de mantener un básico ritmo de trabajo. Estoy consciente de mis valores, fortalezas pero no de mis debilidades con el n de recurrir a la ayuda necesaria de cualquier fuente de manera que conserve el respeto debido a mis compañeros y docente. Reexiono ocasionalmente sobre la aplicación de las integrales en diversas áreas pero no sobre su utilidad en ellas.
Utilizo en ocasiones el diálogo entre mis compañeros y docente para resolver situaciones sencillas presentadas. Mantengo ocasionalmente una actitud positiva frente a las dicultades que se presentan al equipo o de forma individual. Ocasionalmente soy perseverante al momento de emplear las fórmulas de áreas y volúmenes con el n de mantener un buen ritmo de trabajo. Estoy consciente de mis valores, fortalezas pero no de mis debilidades con el n de recurrir en ocasiones a la ayuda necesaria de cualquier fuente de manera que conserve el respeto debido a mis compañeros y docente. Reexiono con errores sobre la aplicación de las integrales en diversas áreas pero no sobre su utilidad en ellas.
No utilizo el diálogo entre mis compañero compañeross y docente para resolver situaciones presentadas. No mantengo una actitud positiva frente a las dicultades que se presentan al equipo o de forma individual. No soy perseverante al momento de emplear las fórmulas de áreas y volúmenes con el n de mantener un buen ritmo de trabajo. No estoy consciente de mis valores, fortalezas ni debilidades. No reexiono sobre la aplicación de las integrales en diversas áreas ni así como de su utilidad en ellas.
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Resuelves problemas de aplicación de la integral definida en situaciones reales en el campo de las ciencias exactas, sociales, naturales y administrativas
NOTAS
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Bloque IV
NOTAS