{"id":1392,"date":"2024-08-23T19:00:00","date_gmt":"2024-08-23T19:00:00","guid":{"rendered":"https:\/\/wuolah.com\/blog\/?p=1392"},"modified":"2024-08-28T18:06:50","modified_gmt":"2024-08-28T18:06:50","slug":"integral-definida","status":"publish","type":"post","link":"https:\/\/wuolah.com\/blog\/integral-definida\/","title":{"rendered":"Integral definida: concepto, propiedades y c\u00e1lculo"},"content":{"rendered":"\n<p><\/p>\n\n\n\n<p>La integral definida tiene m\u00faltiples aplicaciones en las distintas ramas de la ciencia. Sigue leyendo este art\u00edculo que te contamos qu\u00e9 es, sus propiedades y c\u00f3mo calcularlas con un ejemplo pr\u00e1ctico.<\/p>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Tabla de contenidos<\/p>\n<span class=\"ez-toc-title-toggle\"><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/wuolah.com\/blog\/integral-definida\/#%C2%BFQue_es_una_integral_definida\" >\u00bfQu\u00e9 es una integral definida?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/wuolah.com\/blog\/integral-definida\/#Propiedades_de_la_integral_definida\" >Propiedades de la integral definida<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/wuolah.com\/blog\/integral-definida\/#Tabla_de_integrales_definidas\" >Tabla de integrales definidas<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/wuolah.com\/blog\/integral-definida\/#Calculo_de_una_integral_definida\" >C\u00e1lculo de una integral definida<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue_es_una_integral_definida\"><\/span>\u00bfQu\u00e9 es una integral definida?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>La integral definida se utiliza para <strong>encontrar el \u00e1rea bajo una curva en un intervalo cerrado<\/strong> [a,b]en el eje x.<\/p>\n<\/blockquote>\n\n\n\n<p>Matem\u00e1ticamente, la integral definida de una funci\u00f3n<strong> f(x)<\/strong> en el intervalo<strong> [a,b]<\/strong> se representa como:<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" width=\"280\" height=\"115\" src=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image.png\" alt=\"\" class=\"wp-image-1411\" style=\"width:192px;height:auto\" \/><\/figure>\n\n\n\n<p>En esta expresi\u00f3n:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>a<\/strong> y <strong>b<\/strong> son los l\u00edmites de integraci\u00f3n que definen el intervalo sobre el cual se est\u00e1 calculando el \u00e1rea.<\/li>\n\n\n\n<li><strong>f(x)<\/strong> es la funci\u00f3n que se va a integrar.<\/li>\n\n\n\n<li><strong>dx<\/strong> indica que la integraci\u00f3n se realiza respecto a la variable x.<\/li>\n<\/ul>\n\n\n\n<p>Si la curva est\u00e1 por encima del eje <strong>x<\/strong>, el \u00e1rea es positiva y si est\u00e1 por debajo, el \u00e1rea es negativa. Adem\u00e1s la integral definida es capaz de combinar estas \u00e1reas, permitiendo calcular tanto el \u00e1rea total como la acumulaci\u00f3n neta de valores de la funci\u00f3n en ese intervalo.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Propiedades_de_la_integral_definida\"><\/span>Propiedades de la integral definida<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>La integral definida tiene varias propiedades importantes que facilitan su c\u00e1lculo y comprensi\u00f3n, te las mostramos a continuaci\u00f3n:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Linealidad<\/strong>: La integral definida es lineal, lo que significa que si <strong>f(x)<\/strong> y <strong>g(x)<\/strong> son funciones integrables y <strong>c<\/strong> es una constante, entonces:<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img fetchpriority=\"high\" decoding=\"async\" width=\"631\" height=\"85\" src=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-1.png\" alt=\"\" class=\"wp-image-1417\" style=\"width:397px;height:auto\" srcset=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-1.png 631w, https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-1-300x40.png 300w\" sizes=\"(max-width: 631px) 100vw, 631px\" \/><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Propiedad aditiva<\/strong>: Si se divide el intervalo de integraci\u00f3n en dos subintervalos <strong>[a,c]<\/strong> y <strong>[c,b]<\/strong>, la integral sobre el intervalo total <strong>[a,b]<\/strong> es la suma de las integrales sobre los subintervalos:<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" width=\"513\" height=\"96\" src=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-2.png\" alt=\"\" class=\"wp-image-1418\" style=\"width:337px;height:auto\" srcset=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-2.png 513w, https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-2-300x56.png 300w\" sizes=\"(max-width: 513px) 100vw, 513px\" \/><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Cambio de l\u00edmites<\/strong>: Si se invierten los l\u00edmites de integraci\u00f3n, el signo de la integral cambia:<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"364\" height=\"102\" src=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-3.png\" alt=\"\" class=\"wp-image-1419\" style=\"width:234px;height:auto\" srcset=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-3.png 364w, https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-3-300x84.png 300w\" sizes=\"(max-width: 364px) 100vw, 364px\" \/><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Integral de una constante<\/strong>: La integral de una constante <strong>c<\/strong> sobre un intervalo <strong>[a,b]<\/strong> es igual a la constante multiplicada por la longitud del intervalo:<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"268\" height=\"102\" src=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-4.png\" alt=\"\" class=\"wp-image-1420\" style=\"width:166px;height:auto\" \/><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Valor medio del teorema del c\u00e1lculo<\/strong>: Existe un punto <strong>c<\/strong> en el intervalo <strong>[a,b]<\/strong> tal que:<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"337\" height=\"99\" src=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-5.png\" alt=\"\" class=\"wp-image-1421\" style=\"width:201px;height:auto\" srcset=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-5.png 337w, https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-5-300x88.png 300w\" sizes=\"(max-width: 337px) 100vw, 337px\" \/><\/figure>\n\n\n\n<p>Esto significa que el \u00e1rea bajo la curva es igual al valor de la funci\u00f3n en alg\u00fan punto del intervalo, multiplicado por la longitud del intervalo.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Tabla_de_integrales_definidas\"><\/span>Tabla de integrales definidas<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Para facilitar el c\u00e1lculo de integrales definidas, se utilizan tablas que contienen integrales comunes:<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"591\" height=\"375\" src=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-6.png\" alt=\"\" class=\"wp-image-1422\" style=\"width:439px;height:auto\" srcset=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-6.png 591w, https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-6-300x190.png 300w\" sizes=\"(max-width: 591px) 100vw, 591px\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Calculo_de_una_integral_definida\"><\/span>C\u00e1lculo de una integral definida<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Para calcular una integral definida se deben seguir los siguientes pasos:<\/p>\n\n\n\n<p>1. <strong>Encontrar la antiderivada<\/strong>: es decir, hallar la funci\u00f3n primitiva o antiderivada <strong>F(x)<\/strong> de la funci\u00f3n <strong>f(x)<\/strong> o lo que es lo mismo encontrar una funci\u00f3n cuya derivada sea <strong>f(x)<\/strong>.<\/p>\n\n\n\n<p>2. <strong>Aplicar la regla fundamental del c\u00e1lculo<\/strong>: Una vez que obtenida la antiderivada <strong>F(x)<\/strong>, se aplica la regla fundamental del c\u00e1lculo que establece que:<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"367\" height=\"106\" src=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-7.png\" alt=\"\" class=\"wp-image-1423\" style=\"width:257px;height:auto\" srcset=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-7.png 367w, https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-7-300x87.png 300w\" sizes=\"(max-width: 367px) 100vw, 367px\" \/><\/figure>\n\n\n\n<p>Es decir, se eval\u00faa la antiderivada en los l\u00edmites superior e inferior y se resta el valor de F(a)F(a)F(a) del valor de F(b)F(b)F(b).<\/p>\n\n\n\n<p>3. <strong>Simplificar el resultado<\/strong>: Finalmente, se realiza la simplificaci\u00f3n necesaria para obtener el valor de la integral definida.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong>Ejemplo de c\u00e1lculo:<\/strong><\/p>\n\n\n\n<p>Calculemos la integral definida de la funci\u00f3n f(x)=3x<sup>2<\/sup> en el intervalo [1,4]:<\/p>\n\n\n\n<p>Lo primero ser\u00e1 encontrar <strong>la antiderivada<\/strong> de 3x<sup>2<\/sup> que es F(x)=x<sup>3<\/sup>, ya que la derivada de x<sup>3<\/sup> es  3x<sup>2<\/sup>.<\/p>\n\n\n\n<p>Si aplicamos ahora la <strong>regla fundamental del c\u00e1lculo<\/strong>:<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"667\" height=\"82\" src=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-8.png\" alt=\"\" class=\"wp-image-1424\" style=\"width:429px;height:auto\" srcset=\"https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-8.png 667w, https:\/\/wuolah.com\/blog\/wp-content\/uploads\/sites\/2\/2024\/08\/image-8-300x37.png 300w\" sizes=\"(max-width: 667px) 100vw, 667px\" \/><\/figure>\n\n\n\n<p>Concluiremos que el <strong>resultado<\/strong> de la integral definida es 63, que representa el \u00e1rea bajo la curva 3x<sup>2<\/sup> desde x=1 hasta x=4.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>La integral definida tiene m\u00faltiples aplicaciones en las distintas ramas de la ciencia. Sigue leyendo este art\u00edculo que te contamos qu\u00e9 es, sus propiedades y c\u00f3mo calcularlas con un ejemplo pr\u00e1ctico. \u00bfQu\u00e9 es una integral definida? La integral definida se utiliza para encontrar el \u00e1rea bajo una curva en un intervalo cerrado [a,b]en el eje [&hellip;]<\/p>\n","protected":false},"author":14,"featured_media":1413,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16],"tags":[],"class_list":["post-1392","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematicas"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.1 (Yoast SEO v25.6) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Integral definida: concepto, propiedades y c\u00e1lculo | Wuolah<\/title>\n<meta name=\"description\" content=\"Conoce qu\u00e9 son las integrales definidas, sus propiedades y c\u00f3mo calcularlas a trav\u00e9s de un ejercicio pr\u00e1ctico.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/wuolah.com\/blog\/integral-definida\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Integral definida: concepto, propiedades y c\u00e1lculo\" \/>\n<meta property=\"og:description\" content=\"Conoce qu\u00e9 son las integrales definidas, sus propiedades y c\u00f3mo calcularlas a trav\u00e9s de un ejercicio pr\u00e1ctico.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/wuolah.com\/blog\/integral-definida\/\" \/>\n<meta property=\"og:site_name\" content=\"Blog de Educaci\u00f3n - 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