{"id":761,"date":"2025-06-20T17:10:42","date_gmt":"2025-06-20T17:10:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=761"},"modified":"2025-07-18T16:20:39","modified_gmt":"2025-07-18T16:20:39","slug":"error-analysis-in-numerical-integration-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/error-analysis-in-numerical-integration-learn-it-1\/","title":{"raw":"Error Analysis in Numerical Integration: Learn It 1","rendered":"Error Analysis in Numerical Integration: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Calculate how far off your numerical approximation might be from the true value<\/li>\r\n \t<li>Use error-bound formulas to estimate the accuracy of your approximation<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Absolute and Relative Error<\/h2>\r\nWhen you use numerical approximation methods like the midpoint rule or trapezoidal rule to estimate definite integrals, it's crucial to understand how accurate your approximation actually is. This involves calculating two types of error: <strong>absolute error<\/strong> and <strong>relative error<\/strong>.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>absolute and relative error<\/h3>\r\n<p class=\"whitespace-normal break-words\">If [latex]B[\/latex] is your estimate of some quantity having an actual value of [latex]A[\/latex], then:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>Absolute error<\/strong> = [latex]|A - B|[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Relative error<\/strong> = [latex]\\left|\\frac{A - B}{A}\\right| \\cdot 100%[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The relative error expresses the error as a percentage of the true value.<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Why Both Matter<\/strong>: Absolute error tells you the actual difference between your estimate and the true value. Relative error tells you how significant that difference is compared to the size of the true value. A small absolute error might still be a large relative error if the true value is very small.<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165040640350\" data-type=\"problem\">\r\n\r\nCalculate the absolute and relative error in the estimate of [latex]{\\displaystyle\\int }_{0}^{1}{x}^{2}dx[\/latex] using the midpoint rule, found in the example: Using the Midpoint Rule with [latex]{M}_{4}[\/latex].\r\n<p id=\"fs-id1165041770837\">[reveal-answer q=\"44558892\"]Show Solution[\/reveal-answer]<\/p>\r\n\r\n<\/div>\r\n[hidden-answer a=\"44558892\"]\r\n<div id=\"fs-id1165041788187\" data-type=\"solution\">\r\n<p id=\"fs-id1165041788189\">The calculated value is [latex]{\\displaystyle\\int }_{0}^{1}{x}^{2}dx=\\frac{1}{3}[\/latex] and our estimate from the example is [latex]{M}_{4}=\\frac{21}{64}[\/latex]. Thus, the absolute error is given by [latex]|\\left(\\frac{1}{3}\\right)-\\left(\\frac{21}{64}\\right)|=\\frac{1}{192}\\approx 0.0052[\/latex]. The relative error is<\/p>\r\n\r\n<div id=\"fs-id1165042048763\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\frac{\\frac{1}{192}}{\\frac{1}{3}}=\\frac{1}{64}\\approx 0.015625\\approx 1.6\\text{%}[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1165040771110\" data-type=\"problem\">\r\n<p id=\"fs-id1165040771115\">Calculate the absolute and relative error in the estimate of [latex]{\\displaystyle\\int }_{0}^{1}{x}^{2}dx[\/latex] using the trapezoidal rule, found in the example: Using the trapezoidal rule.<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558891\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558891\"]\r\n<div id=\"fs-id1165042091200\" data-type=\"solution\">\r\n<p id=\"fs-id1165042091202\">The calculated value is [latex]{\\displaystyle\\int }_{0}^{1}{x}^{2}dx=\\frac{1}{3}[\/latex] and our estimate from the example is [latex]{T}_{4}=\\frac{11}{32}[\/latex]. Thus, the absolute error is given by [latex]|\\frac{1}{3}-\\frac{11}{32}|=\\frac{1}{96}\\approx 0.0104[\/latex]. The relative error is given by<\/p>\r\n\r\n<div id=\"fs-id1165040745022\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\frac{\\frac{1}{96}}{\\frac{1}{3}}=0.03125\\approx 3.1\\text{%}[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solutions to the example above.<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=6722688&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=onI9vFqmDQU&amp;video_target=tpm-plugin-zltaqp36-onI9vFqmDQU\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/center>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.6.2_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for \"3.6.2\" here (opens in new window)<\/a>.\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Calculate how far off your numerical approximation might be from the true value<\/li>\n<li>Use error-bound formulas to estimate the accuracy of your approximation<\/li>\n<\/ul>\n<\/section>\n<h2>Absolute and Relative Error<\/h2>\n<p>When you use numerical approximation methods like the midpoint rule or trapezoidal rule to estimate definite integrals, it&#8217;s crucial to understand how accurate your approximation actually is. This involves calculating two types of error: <strong>absolute error<\/strong> and <strong>relative error<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>absolute and relative error<\/h3>\n<p class=\"whitespace-normal break-words\">If [latex]B[\/latex] is your estimate of some quantity having an actual value of [latex]A[\/latex], then:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>Absolute error<\/strong> = [latex]|A - B|[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Relative error<\/strong> = [latex]\\left|\\frac{A - B}{A}\\right| \\cdot 100%[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The relative error expresses the error as a percentage of the true value.<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\"><strong>Why Both Matter<\/strong>: Absolute error tells you the actual difference between your estimate and the true value. Relative error tells you how significant that difference is compared to the size of the true value. A small absolute error might still be a large relative error if the true value is very small.<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165040640350\" data-type=\"problem\">\n<p>Calculate the absolute and relative error in the estimate of [latex]{\\displaystyle\\int }_{0}^{1}{x}^{2}dx[\/latex] using the midpoint rule, found in the example: Using the Midpoint Rule with [latex]{M}_{4}[\/latex].<\/p>\n<p id=\"fs-id1165041770837\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558892\">Show Solution<\/button><\/p>\n<\/div>\n<div id=\"q44558892\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165041788187\" data-type=\"solution\">\n<p id=\"fs-id1165041788189\">The calculated value is [latex]{\\displaystyle\\int }_{0}^{1}{x}^{2}dx=\\frac{1}{3}[\/latex] and our estimate from the example is [latex]{M}_{4}=\\frac{21}{64}[\/latex]. Thus, the absolute error is given by [latex]|\\left(\\frac{1}{3}\\right)-\\left(\\frac{21}{64}\\right)|=\\frac{1}{192}\\approx 0.0052[\/latex]. The relative error is<\/p>\n<div id=\"fs-id1165042048763\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\frac{\\frac{1}{192}}{\\frac{1}{3}}=\\frac{1}{64}\\approx 0.015625\\approx 1.6\\text{%}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1165040771110\" data-type=\"problem\">\n<p id=\"fs-id1165040771115\">Calculate the absolute and relative error in the estimate of [latex]{\\displaystyle\\int }_{0}^{1}{x}^{2}dx[\/latex] using the trapezoidal rule, found in the example: Using the trapezoidal rule.<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558891\">Show Solution<\/button><\/p>\n<div id=\"q44558891\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165042091200\" data-type=\"solution\">\n<p id=\"fs-id1165042091202\">The calculated value is [latex]{\\displaystyle\\int }_{0}^{1}{x}^{2}dx=\\frac{1}{3}[\/latex] and our estimate from the example is [latex]{T}_{4}=\\frac{11}{32}[\/latex]. Thus, the absolute error is given by [latex]|\\frac{1}{3}-\\frac{11}{32}|=\\frac{1}{96}\\approx 0.0104[\/latex]. The relative error is given by<\/p>\n<div id=\"fs-id1165040745022\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\frac{\\frac{1}{96}}{\\frac{1}{3}}=0.03125\\approx 3.1\\text{%}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solutions to the example above.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=6722688&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=onI9vFqmDQU&amp;video_target=tpm-plugin-zltaqp36-onI9vFqmDQU\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/3.6.2_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;3.6.2&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":11,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":667,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/761"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/761\/revisions"}],"predecessor-version":[{"id":1328,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/761\/revisions\/1328"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/667"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/761\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=761"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=761"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=761"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=761"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}