{"id":26655,"date":"2023-08-01T19:26:30","date_gmt":"2023-08-01T19:26:30","guid":{"rendered":"https:\/\/blogs.ugto.mx\/rea\/?p=26655"},"modified":"2023-08-30T15:00:23","modified_gmt":"2023-08-30T15:00:23","slug":"clase-digital-10-area-bajo-la-curva","status":"publish","type":"post","link":"https:\/\/blogs.ugto.mx\/rea\/clase-digital-10-area-bajo-la-curva\/","title":{"rendered":"Clase digital 10. \u00c1rea bajo la curva"},"content":{"rendered":"\n\n\n<div class=\"wp-block-cover is-light\" style=\"min-height:284px;aspect-ratio:unset;\"><span aria-hidden=\"true\" class=\"wp-block-cover__background has-background-dim-40 has-background-dim\"><\/span><img decoding=\"async\" class=\"wp-block-cover__image-background wp-image-26657\" alt=\"compass, direction, clock\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/73112.jpg\" style=\"object-position:77% 82%\" data-object-fit=\"cover\" data-object-position=\"77% 82%\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"1280\" height=\"827\" class=\"wp-block-cover__image-background wp-image-26657\" alt=\"compass, direction, clock\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/73112.jpg\" style=\"object-position:77% 82%\" data-object-fit=\"cover\" data-object-position=\"77% 82%\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/73112.jpg 1280w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/73112-300x194.jpg 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/73112-1024x662.jpg 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/73112-768x496.jpg 768w\" sizes=\"auto, (max-width: 1280px) 100vw, 1280px\" \/><\/noscript><div class=\"wp-block-cover__inner-container is-layout-flow wp-block-cover-is-layout-flow\">\n<p class=\"has-text-align-center has-base-3-color has-text-color has-large-font-size wp-block-paragraph\">\u00c1rea bajo la curva<\/p>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"introduccion\">Introducci\u00f3n<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Una de las aplicaciones m\u00e1s simples de una integral es el <strong>\u00e1rea de una regi\u00f3n plana bajo una curva<\/strong>. Para ello primero veremos el concepto de sumatoria, despu\u00e9s dos definiciones simples y finalmente ejemplos de aplicaci\u00f3n.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"desarrollo-del-tema\">Desarrollo del tema <\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u00c1rea bajo la curva<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Hemos llegado al final del curso, ahora veremos una de las aplicaciones de la integral m\u00e1s com\u00fan, el c\u00e1lculo del \u00e1rea bajo una curva. Para ello primero deberemos conocer lo que es <strong>una serie<\/strong>, <strong>una sumatoria <\/strong>y finalmente <strong>una suma de Riemann<\/strong>. Supongamos que deseas sumar muchos n\u00fameros, por ejemplo: del 1 al 100. Entonces escribir\u00edas:<\/p>\n\n\n\n<p class=\"has-text-align-center has-medium-font-size wp-block-paragraph\"><strong>1+2+3+4+5+\u2026.+99+100<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Pero hay una forma m\u00e1s corta de escribirlo que nos ayuda mucho, usando un s\u00edmbolo que significa sumatoria , es una letra griega de nombre <strong>sigma<\/strong> <strong>may\u00fascula<\/strong>. Adem\u00e1s usamos una letra que nos indica cu\u00e1ntos vamos a sumar, denotada por <strong><em>i<\/em><\/strong>, as\u00ed nuestra suma quedar\u00eda expresada como:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.29.35.png\" alt=\"\" class=\"wp-image-28301\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"84\" height=\"108\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.29.35.png\" alt=\"\" class=\"wp-image-28301\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">De esa manera podemos expresar cualquier suma de n\u00fameros o funciones de n\u00fameros que tengan un orden, por ejemplo:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.30.03.png\" alt=\"\" class=\"wp-image-28303\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"868\" height=\"280\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.30.03.png\" alt=\"\" class=\"wp-image-28303\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.30.03.png 868w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.30.03-300x97.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.30.03-768x248.png 768w\" sizes=\"auto, (max-width: 868px) 100vw, 868px\" \/><\/noscript><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Existen teoremas que nos permiten simplificar estas sumatorias, te pedir\u00e9 que los revises en la secci\u00f3n 4.4 del libro de <strong>Leithold<\/strong>. Con esto en mente podemos calcular el \u00e1rea de cualquier figura como la suma de sus \u00e1reas, por ejemplo, en la siguiente figura podemos calcular el \u00e1rea de la regi\u00f3n limitada por las l\u00edneas negras como una suma de las \u00e1reas marcadas con rojo:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.35.08.png\" alt=\"\" class=\"wp-image-28304\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"432\" height=\"380\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.35.08.png\" alt=\"\" class=\"wp-image-28304\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.35.08.png 432w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.35.08-300x264.png 300w\" sizes=\"auto, (max-width: 432px) 100vw, 432px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">O tal vez, esta, donde aproximamos el \u00e1rea bajo la curva roja, tomando el \u00e1rea de varios rect\u00e1ngulos (verdes) y sum\u00e1ndolos:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.37.06.png\" alt=\"\" class=\"wp-image-28305\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"609\" height=\"1025\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.37.06.png\" alt=\"\" class=\"wp-image-28305\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.37.06.png 609w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.37.06-178x300.png 178w\" sizes=\"auto, (max-width: 609px) 100vw, 609px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Ahora pasemos a la definici\u00f3n del \u00e1rea de una regi\u00f3n plana. Suponga que la funci\u00f3n <strong><em>f<\/em><\/strong> es continua en el intervalo <strong><em>cerrado<\/em><\/strong><em> <\/em><strong><em>[a,b]<\/em><\/strong><strong> <\/strong>con <strong><em>f<\/em><\/strong><strong>(x)\u22650<\/strong>&nbsp; para toda <strong><em>x<\/em><\/strong> en el intervalo<strong> <\/strong><strong><em>[a,b]<\/em><\/strong><strong> <\/strong>, y que <strong><em>R<\/em><\/strong> es la regi\u00f3n limitada por la curva <strong>&nbsp;<\/strong><strong><em>y=f<\/em><\/strong><strong>(x)<\/strong>, el eje <strong><em>X<\/em><\/strong> y las rectas <strong><em>x=a<\/em><\/strong> y <strong><em>x=b<\/em><\/strong>. Divide el intervalo<strong> <\/strong><strong><em>[a,b]<\/em><\/strong><strong> <\/strong>en <strong><em>n<\/em><\/strong><em> <\/em>&nbsp;subintervalos, cada uno de longitud <strong>\u0394<\/strong><strong><em>x=(b-a)\/n<\/em><\/strong>, y denote el i-\u00e9simo subintervalo<strong> <\/strong><strong><em>[x<\/em><\/strong><strong><em><sub>i-1<\/sub><\/em><\/strong><strong><em>,x<\/em><\/strong><strong><em><sub>i<\/sub><\/em><\/strong><strong><em>]<\/em><\/strong><em>.<\/em> entonces si <strong><em>f<\/em><\/strong><strong>(<\/strong><strong><em>c<\/em><\/strong><strong><em><sub>i<\/sub><\/em><\/strong><strong>)<\/strong> es el valor de la funci\u00f3n m\u00ednimo absoluto en el i-\u00e9simo subintervalo, la medida del \u00e1rea de la regi\u00f3n <strong><em>R<\/em><\/strong> est\u00e1 dada por<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.37.41.png\" alt=\"\" class=\"wp-image-28306\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"214\" height=\"101\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.37.41.png\" alt=\"\" class=\"wp-image-28306\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Veamos algunos ejemplos de c\u00f3mo aplicar esta suma de \u00e1reas y los teoremas que te ped\u00ed revisaras.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ejemplo:<br>Calcule el \u00e1rea de la regi\u00f3n limitada por<strong> <em>y = x<sup>2<\/sup><\/em><\/strong>, el eje <strong><em>x<\/em><\/strong>, y la recta <strong><em>x = 2<\/em><\/strong>, usando a) rect\u00e1ngulos inscritos, b) rect\u00e1ngulos circunscritos. Grafiquemos primero para darnos una idea de que \u00e1rea nos piden:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.38.15-1024x466.png\" alt=\"\" class=\"wp-image-28307\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"466\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.38.15-1024x466.png\" alt=\"\" class=\"wp-image-28307\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.38.15-1024x466.png 1024w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.38.15-300x137.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.38.15-768x350.png 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.38.15.png 1096w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">En la figura <strong>A<\/strong> vemos en verde el \u00e1rea que nos piden y en la figura <strong>B <\/strong>se muestran unos rect\u00e1ngulos inscritos como ejemplo.\u00a0 Comencemos por calcular el \u00e1rea usando solo una sumatoria sin aplicar la definici\u00f3n de \u00e1rea de una regi\u00f3n plana. Bien supongamos que dividimos el intervalo de <strong>x=0 <\/strong>a <strong>x=2<\/strong> en 10 partes, as\u00ed que <strong>2-0\/10=0.2<\/strong> el <strong>\u0394x =0.2<\/strong>:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.39.19.png\" alt=\"\" class=\"wp-image-28308\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"924\" height=\"768\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.39.19.png\" alt=\"\" class=\"wp-image-28308\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.39.19.png 924w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.39.19-300x249.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.39.19-768x638.png 768w\" sizes=\"auto, (max-width: 924px) 100vw, 924px\" \/><\/noscript><\/figure>\n<\/div>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.39.43.png\" alt=\"\" class=\"wp-image-28309\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"866\" height=\"552\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.39.43.png\" alt=\"\" class=\"wp-image-28309\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.39.43.png 866w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.39.43-300x191.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.39.43-768x490.png 768w\" sizes=\"auto, (max-width: 866px) 100vw, 866px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Si comparamos este resultado con el anterior veremos que el primero es menor, \u00bfpor qu\u00e9? Porque hay espacios que no se consideraron (ver la figura b en color amarillo).\u00a0 Ahora consideremos rect\u00e1ngulos circunscritos.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.40.13.png\" alt=\"\" class=\"wp-image-28310\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"422\" height=\"386\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.40.13.png\" alt=\"\" class=\"wp-image-28310\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.40.13.png 422w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.40.13-300x274.png 300w\" sizes=\"auto, (max-width: 422px) 100vw, 422px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<p class=\"wp-block-paragraph\">Comencemos por calcular el \u00e1rea usando solo una sumatoria sin aplicar la definici\u00f3n de \u00e1rea de una regi\u00f3n plana. Bien supongamos que dividimos el intervalo de <strong>x = 0<\/strong> a <strong>x = 2<\/strong> en 10 partes, as\u00ed que <strong>2-0\/10 = 0.2<\/strong> el <strong>\u0394x  = 0.2<\/strong>.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.41.04.png\" alt=\"\" class=\"wp-image-28311\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"886\" height=\"730\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.41.04.png\" alt=\"\" class=\"wp-image-28311\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.41.04.png 886w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.41.04-300x247.png 300w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.41.04-768x633.png 768w\" sizes=\"auto, (max-width: 886px) 100vw, 886px\" \/><\/noscript><\/figure>\n<\/div>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.41.36-907x1024.png\" alt=\"\" class=\"wp-image-28312\" \/><noscript><img loading=\"lazy\" decoding=\"async\" width=\"907\" height=\"1024\" src=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.41.36-907x1024.png\" alt=\"\" class=\"wp-image-28312\" srcset=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.41.36-907x1024.png 907w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.41.36-266x300.png 266w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.41.36-768x867.png 768w, https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/Captura-de-pantalla-2023-08-30-a-las-8.41.36.png 934w\" sizes=\"auto, (max-width: 907px) 100vw, 907px\" \/><\/noscript><\/figure>\n<\/div>\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Nombre<\/th><th class=\"has-text-align-center\" data-align=\"center\">Enlace<\/th><\/tr><\/thead><tbody><tr><td>\u00c1rea bajo una curva &#8211; Ejercicio 1<\/td><td class=\"has-text-align-center\" data-align=\"center\">[<a href=\"https:\/\/www.youtube.com\/watch?v=V7WnsXYJZaM\" target=\"_blank\" rel=\"noreferrer noopener\">Acceder<\/a>]<\/td><\/tr><tr><td>\u00c1rea bajo una curva &#8211; Ejercicio 2<\/td><td class=\"has-text-align-center\" data-align=\"center\">[<a href=\"https:\/\/www.youtube.com\/watch?v=8tYaZw-1vD4\" target=\"_blank\" rel=\"noreferrer noopener\">Acceder<\/a>]<\/td><\/tr><tr><td>\u00c1rea bajo la curva | Introducci\u00f3n<\/td><td class=\"has-text-align-center\" data-align=\"center\">[<a href=\"https:\/\/www.youtube.com\/watch?v=5ZrfmQEVMjk\" target=\"_blank\" rel=\"noreferrer noopener\">Acceder<\/a>]<\/td><\/tr><tr><td>\u00c1rea bajo la curva | Ejemplo 2<\/td><td class=\"has-text-align-center\" data-align=\"center\">[<a href=\"https:\/\/www.youtube.com\/watch?v=JfwLXVwPpkU\" target=\"_blank\" rel=\"noreferrer noopener\">Acceder<\/a>]<\/td><\/tr><tr><td>\u00c1rea bajo la curva | Ejemplo 3<\/td><td class=\"has-text-align-center\" data-align=\"center\">[<a href=\"https:\/\/www.youtube.com\/watch?v=SDyyHsKXx04\" target=\"_blank\" rel=\"noreferrer noopener\">Acceder<\/a>]<\/td><\/tr><tr><td>\u00c1rea bajo la curva | Ejemplo 4<\/td><td class=\"has-text-align-center\" data-align=\"center\">[<a href=\"https:\/\/www.youtube.com\/watch?v=qv4PWYWVg6s\" target=\"_blank\" rel=\"noreferrer noopener\">Acceder<\/a>]<\/td><\/tr><tr><td>\u00c1rea bajo la curva | Ejemplo 5<\/td><td class=\"has-text-align-center\" data-align=\"center\">[<a href=\"https:\/\/www.youtube.com\/watch?v=sa77YylT0q0\" target=\"_blank\" rel=\"noreferrer noopener\">Acceder<\/a>]<\/td><\/tr><tr><td>\u00c1rea bajo la curva\u2502ejercicio 1<\/td><td class=\"has-text-align-center\" data-align=\"center\">[<a href=\"https:\/\/www.youtube.com\/watch?v=LWq_OV6JAMM\" target=\"_blank\" rel=\"noreferrer noopener\">Acceder<\/a>]<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Te recomiendo que leas los siguientes textos, en donde se ahonda m\u00e1s en el tema y hay m\u00e1s ejemplos.<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/calculo-louis-leithold.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Incrustado de calculo-louis-leithold.\"><\/object><a id=\"wp-block-file--media-6133e632-58d1-4411-8616-ccf80abab537\" href=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/calculo-louis-leithold.pdf\">calculo-louis-leithold<\/a><a href=\"https:\/\/blogs.ugto.mx\/rea\/wp-content\/uploads\/sites\/71\/2023\/08\/calculo-louis-leithold.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-6133e632-58d1-4411-8616-ccf80abab537\">Descarga<\/a><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Una vez que ha concluido con el estudio de material escrito y video, realiza la consigna.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"conclusion\">Conclusi\u00f3n<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Bien hecho, hemos llegado al final del curso, ahora ya tienes los conocimientos b\u00e1sicos para poder entender muchas de las aplicaciones de la derivada y la integral que ver\u00e1s en tus otros cursos.<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\u00a1Felicidades!<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"fuentes-de-informacion\">Fuentes de informaci\u00f3n<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/luiscastellanos.files.wordpress.com\/2007\/02\/calculo-louis-leithold.pdf\">https:\/\/luiscastellanos.files.wordpress.com\/2007\/02\/calculo-louis-leithold.pdf<\/a>, p\u00e1ginas 328-352 numeraci\u00f3n del libro<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Introducci\u00f3n Una de las aplicaciones m\u00e1s simples de una integral es el \u00e1rea de una regi\u00f3n plana bajo una curva. Para ello primero veremos el concepto de sumatoria, despu\u00e9s dos definiciones simples y finalmente ejemplos de aplicaci\u00f3n. Desarrollo del tema \u00c1rea bajo la curva Hemos llegado al final del curso, ahora veremos una de las &#8230; <a title=\"Clase digital 10. \u00c1rea bajo la curva\" class=\"read-more\" href=\"https:\/\/blogs.ugto.mx\/rea\/clase-digital-10-area-bajo-la-curva\/\" aria-label=\"Leer m\u00e1s sobre Clase digital 10. \u00c1rea bajo la curva\">Leer m\u00e1s<\/a><\/p>\n","protected":false},"author":142,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_crdt_document":"","episode_type":"","audio_file":"","podmotor_file_id":"","podmotor_episode_id":"","cover_image":"","cover_image_id":"","duration":"","filesize":"","filesize_raw":"","date_recorded":"","explicit":"","block":"","itunes_episode_number":"","itunes_title":"","itunes_season_number":"","itunes_episode_type":"","footnotes":""},"categories":[19,685],"tags":[41,176,64],"class_list":["post-26655","post","type-post","status-publish","format-standard","hentry","category-ingenieria-en-sistemas-computacionales","category-uda-calculo-diferencial-ingenieria-en-sistemas-computacionales","tag-clase-digital","tag-monica-trejo-duran","tag-neli06002"],"acf":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/26655","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/users\/142"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/comments?post=26655"}],"version-history":[{"count":4,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/26655\/revisions"}],"predecessor-version":[{"id":28315,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/posts\/26655\/revisions\/28315"}],"wp:attachment":[{"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/media?parent=26655"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/categories?post=26655"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ugto.mx\/rea\/wp-json\/wp\/v2\/tags?post=26655"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}