{"id":4201,"date":"2025-07-11T01:14:49","date_gmt":"2025-07-11T01:14:49","guid":{"rendered":"https:\/\/lash.utrng.edu.mx\/?p=4201"},"modified":"2025-07-11T13:48:28","modified_gmt":"2025-07-11T13:48:28","slug":"la-integral-definida","status":"publish","type":"post","link":"https:\/\/lash.utrng.edu.mx\/?p=4201","title":{"rendered":"02. La integral definida"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote has-text-align-center is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-large-font-size\"><em>El objetivo principal del c\u00e1lculo integral es obtener el l\u00edmite de la suma de un gran n\u00famero de magnitudes, cada una de las cuales tiende a cero<\/em><\/p>\n<\/blockquote>\n\n\n\n<p>El fil\u00f3sofo <strong>Brison<\/strong>, contempor\u00e1neo de <strong>S\u00f3crates<\/strong>, trat\u00f3 de calcular el \u00e1rea de un c\u00edrculo por medio de pol\u00edgonos regulares <strong>inscritos<\/strong> y <strong>circunscritos <\/strong>al c\u00edrculo. Este legendario m\u00e9todo se conoce como <strong>proceso de reducci\u00f3n<\/strong> porque a <em>medida que el n\u00famero de lados de un pol\u00edgono aumenta, la diferencia entre las \u00e1reas de \u00e9stos se va reduciendo<\/em>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1015\" height=\"347\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image.png\" alt=\"\" class=\"wp-image-4202\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image.png 1015w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-300x103.png 300w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-768x263.png 768w\" sizes=\"auto, (max-width: 1015px) 100vw, 1015px\" \/><\/figure>\n\n\n\n<!--more-->\n\n\n\n<p>La importancia de esta t\u00e9cnica la podemos observar si se plantea el problema siguiente:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Calcular el \u00e1rea A de la superficie limitada por la par\u00e1bola y = x<sup>2<\/sup> + 1 y las rectas y = 1, x = 0, x = 4<\/li>\n<\/ul>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"616\" height=\"534\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-1.png\" alt=\"\" class=\"wp-image-4203\" style=\"width:692px;height:auto\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-1.png 616w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-1-300x260.png 300w\" sizes=\"auto, (max-width: 616px) 100vw, 616px\" \/><figcaption class=\"wp-element-caption\">Si se divide el segmento 0 a 4 en partes iguales y se trazan dos series de rect\u00e1ngulos, unos tocar\u00e1n la curva con v\u00e9rtice superior izquierdo y los otros la tocar\u00e1n con el v\u00e9rtice superior derecho(fig. 2 y 3)<\/figcaption><\/figure><\/div>\n\n\n<p>La suma de las \u00e1reas de las dos series de rect\u00e1ngulos se presenta a continuaci\u00f3n. En la figura 2 todos los rect\u00e1ngulos tienen base 1:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"683\" height=\"181\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-2.png\" alt=\"\" class=\"wp-image-4204\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-2.png 683w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-2-300x80.png 300w\" sizes=\"auto, (max-width: 683px) 100vw, 683px\" \/><\/figure>\n\n\n\n<p>En la figura 3 todos los rect\u00e1ngulos tienen base 1:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"686\" height=\"193\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-3.png\" alt=\"\" class=\"wp-image-4205\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-3.png 686w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-3-300x84.png 300w\" sizes=\"auto, (max-width: 686px) 100vw, 686px\" \/><\/figure>\n\n\n\n<p>El \u00e1rea para obtener est\u00e1 en 18 y 34 unidades cuadradas: <strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-luminous-vivid-orange-color\">18 &lt; A &lt; 34<\/mark><\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color has-link-color wp-elements-9180a556d5805a41f4e96b037bc74a55\">Suma de Riemann<\/h3>\n\n\n\n<p>La sumatoria de <em>n<\/em> t\u00e9rminos {a<sub>1<\/sub>, a<sub>2<\/sub>, a<sub>3<\/sub>,\u2026a<sub>n<\/sub>} se expresa as\u00ed:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"376\" height=\"86\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-4.png\" alt=\"\" class=\"wp-image-4206\" style=\"width:394px;height:auto\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-4.png 376w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-4-300x69.png 300w\" sizes=\"auto, (max-width: 376px) 100vw, 376px\" \/><\/figure><\/div>\n\n\n<p>De donde:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1010\" height=\"234\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-5.png\" alt=\"\" class=\"wp-image-4207\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-5.png 1010w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-5-300x70.png 300w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-5-768x178.png 768w\" sizes=\"auto, (max-width: 1010px) 100vw, 1010px\" \/><\/figure>\n\n\n\n<p class=\"has-text-align-right\">**Algunos autores usan la palabra <strong><em>l\u00edmite <\/em><\/strong>en lugar de <strong><em>extremos<\/em><\/strong>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color has-link-color wp-elements-c722af0763bcf0a7f0beaaef010fac23\">Ejemplo 1:<\/h3>\n\n\n\n<p>Calcula la siguiente sumatoria:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"146\" height=\"88\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-6.png\" alt=\"\" class=\"wp-image-4208\" style=\"width:194px;height:auto\"\/><\/figure><\/div>\n\n\n<p>En este ejemplo, a<sub>i<\/sub> = (2<sub>i<\/sub> + 1). Para calcular la suma sustituimos la<strong><em> i<\/em><\/strong> sucesivamente por los enteros 1, 2, 3, 4 desde el 1 hasta el 4, que en el ejemplo son los externos de la sumatoria, luego se suman los t\u00e9rminos as\u00ed obtenidos:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"797\" height=\"209\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-7.png\" alt=\"\" class=\"wp-image-4209\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-7.png 797w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-7-300x79.png 300w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-7-768x201.png 768w\" sizes=\"auto, (max-width: 797px) 100vw, 797px\" \/><\/figure>\n\n\n\n<p>Cualquier variable se puede usar como \u00edndice de la sumatoria. Sin embargo, es muy com\u00fan preferir las letras i, j, k porque normalmente est\u00e1n asociadas con los enteros. El extremo inferior no tiene que ser necesariamente el n\u00famero 1, pues cualquier n\u00famero entero menor o igual al extremo superior es v\u00e1lido.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"369\" height=\"88\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-8.png\" alt=\"\" class=\"wp-image-4210\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-8.png 369w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-8-300x72.png 300w\" sizes=\"auto, (max-width: 369px) 100vw, 369px\" \/><\/figure><\/div>\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color has-link-color wp-elements-1eaa52d1b943ebc54aa371ee919bafe6\">Ejemplo 2:<\/h3>\n\n\n\n<p>Calcula las siguientes suma:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"794\" height=\"191\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-9.png\" alt=\"\" class=\"wp-image-4211\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-9.png 794w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-9-300x72.png 300w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-9-768x185.png 768w\" sizes=\"auto, (max-width: 794px) 100vw, 794px\" \/><\/figure>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"295\" height=\"86\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-10.png\" alt=\"\" class=\"wp-image-4212\"\/><\/figure><\/div>\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"765\" height=\"248\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-11.png\" alt=\"\" class=\"wp-image-4213\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-11.png 765w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-11-300x97.png 300w\" sizes=\"auto, (max-width: 765px) 100vw, 765px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color has-link-color wp-elements-6aebbe810f597a7defb4502a30364a0b\">Propiedades de las sumas de Riemann<\/h2>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"616\" height=\"225\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-12.png\" alt=\"\" class=\"wp-image-4214\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-12.png 616w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-12-300x110.png 300w\" sizes=\"auto, (max-width: 616px) 100vw, 616px\" \/><\/figure>\n\n\n\n<p class=\"has-vivid-cyan-blue-color has-text-color has-link-color wp-elements-d4c86fb26d69a995ebb17f6a369bf8e9\"><strong>Ejemplo de las propiedades:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"552\" height=\"638\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-13.png\" alt=\"\" class=\"wp-image-4215\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-13.png 552w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-13-260x300.png 260w\" sizes=\"auto, (max-width: 552px) 100vw, 552px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color has-link-color wp-elements-f5ec11d043362234170378b00450132a\">F\u00f3rmulas de las sumas de Riemann<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"530\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-14-1024x530.png\" alt=\"\" class=\"wp-image-4216\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-14-1024x530.png 1024w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-14-300x155.png 300w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-14-768x398.png 768w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-14.png 1062w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-white-color has-vivid-cyan-blue-background-color has-text-color has-background has-link-color wp-elements-18d28b137fbfeb53f32f578543d6ddad\">Ejercicios:<\/h3>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"184\" height=\"95\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-15.png\" alt=\"\" class=\"wp-image-4217\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"173\" height=\"88\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-16.png\" alt=\"\" class=\"wp-image-4218\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"99\" height=\"90\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-17.png\" alt=\"\" class=\"wp-image-4219\"\/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"351\" height=\"96\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-18.png\" alt=\"\" class=\"wp-image-4220\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-18.png 351w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-18-300x82.png 300w\" sizes=\"auto, (max-width: 351px) 100vw, 351px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-af6a47815711c4dddc6ce094961754e1\">Propiedades de la integral definida<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">A. <\/h3>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"381\" height=\"69\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-19.png\" alt=\"\" class=\"wp-image-4225\" style=\"width:436px;height:auto\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-19.png 381w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-19-300x54.png 300w\" sizes=\"auto, (max-width: 381px) 100vw, 381px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">B.<\/h3>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"633\" height=\"72\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-20.png\" alt=\"\" class=\"wp-image-4226\" style=\"width:703px;height:auto\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-20.png 633w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-20-300x34.png 300w\" sizes=\"auto, (max-width: 633px) 100vw, 633px\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-2817b08d233ede065f552ecf248b4c7f\">Ejemplos de las propiedades A y B<\/h3>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"798\" height=\"77\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-21.png\" alt=\"\" class=\"wp-image-4227\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-21.png 798w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-21-300x29.png 300w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-21-768x74.png 768w\" sizes=\"auto, (max-width: 798px) 100vw, 798px\" \/><\/figure>\n\n\n\n<p>Integramos por separado cada integral:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"642\" height=\"304\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-22.png\" alt=\"\" class=\"wp-image-4228\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-22.png 642w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-22-300x142.png 300w\" sizes=\"auto, (max-width: 642px) 100vw, 642px\" \/><\/figure>\n\n\n\n<p>Por lo tanto,<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"741\" height=\"362\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-23.png\" alt=\"\" class=\"wp-image-4229\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-23.png 741w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-23-300x147.png 300w\" sizes=\"auto, (max-width: 741px) 100vw, 741px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"776\" height=\"68\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-24.png\" alt=\"\" class=\"wp-image-4230\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-24.png 776w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-24-300x26.png 300w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-24-768x67.png 768w\" sizes=\"auto, (max-width: 776px) 100vw, 776px\" \/><\/figure>\n\n\n\n<p>Integramos por separado cada integral:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"703\" height=\"294\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-25.png\" alt=\"\" class=\"wp-image-4231\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-25.png 703w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-25-300x125.png 300w\" sizes=\"auto, (max-width: 703px) 100vw, 703px\" \/><\/figure>\n\n\n\n<p>Por lo tanto,<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"733\" height=\"186\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-26.png\" alt=\"\" class=\"wp-image-4232\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-26.png 733w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-26-300x76.png 300w\" sizes=\"auto, (max-width: 733px) 100vw, 733px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading has-white-color has-vivid-purple-background-color has-text-color has-background has-link-color wp-elements-783bd5b3e139773fd0105637739be9b6\">Ejercicios:<\/h3>\n\n\n\n<h3 class=\"wp-block-heading\">A.<\/h3>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"186\" height=\"73\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-27.png\" alt=\"\" class=\"wp-image-4235\" style=\"width:198px;height:auto\"\/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">B.<\/h3>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"128\" height=\"75\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-28.png\" alt=\"\" class=\"wp-image-4236\" style=\"width:165px;height:auto\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading has-vivid-red-color has-text-color has-link-color wp-elements-6d4364d1aa3ae91878bb984c135552f0\">Teorema fundamental del c\u00e1lculo<\/h2>\n\n\n\n<p>El teorema fundamental del c\u00e1lculo establece que si una funci\u00f3n<em> f<\/em> es continua en el intervalo [a, b], entonces:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"372\" height=\"71\" src=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-29.png\" alt=\"\" class=\"wp-image-4239\" style=\"width:487px;height:auto\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-29.png 372w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-29-300x57.png 300w\" sizes=\"auto, (max-width: 372px) 100vw, 372px\" \/><\/figure><\/div>\n\n\n<p>donde <em>F<\/em> es cualquier funci\u00f3n tal que <em>F<\/em>=<img loading=\"lazy\" decoding=\"async\" width=\"115\" height=\"69\" class=\"wp-image-4240\" style=\"width: 115px;\" src=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-30.png\" alt=\"\">para toda x en [a, b].<\/p>\n\n\n\n<h2 class=\"wp-block-heading has-vivid-red-color has-text-color has-link-color wp-elements-6f1e0dd1944868e78823aaa85b0dd31a\">\u00c1reas<\/h2>\n\n\n\n<p>Por el teorema fundamental de c\u00e1lculo sabemos que si f es una funci\u00f3n continua en el intervalo [a, b], entonces existe la integral definida <img loading=\"lazy\" decoding=\"async\" width=\"115\" height=\"69\" class=\"wp-image-4240\" style=\"width: 115px;\" src=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2025\/07\/image-30.png\" alt=\"\">dx. El resultado de esta integral es igual al \u00e1rea bajo la curva f (x) representada en el plano.<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>El objetivo principal del c\u00e1lculo integral es obtener el l\u00edmite de la suma de un gran n\u00famero de magnitudes, cada una de las cuales tiende a cero El fil\u00f3sofo Brison, contempor\u00e1neo de S\u00f3crates, trat\u00f3 de calcular el \u00e1rea de un c\u00edrculo por medio de pol\u00edgonos regulares inscritos y circunscritos al c\u00edrculo. Este legendario m\u00e9todo se &hellip; <a href=\"https:\/\/lash.utrng.edu.mx\/?p=4201\">Seguir leyendo <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[99],"tags":[],"class_list":["post-4201","post","type-post","status-publish","format-standard","hentry","category-calcintegral-primparc"],"_links":{"self":[{"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/posts\/4201","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4201"}],"version-history":[{"count":8,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/posts\/4201\/revisions"}],"predecessor-version":[{"id":4284,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/posts\/4201\/revisions\/4284"}],"wp:attachment":[{"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4201"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4201"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4201"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}