{"id":753,"date":"2025-06-20T17:10:09","date_gmt":"2025-06-20T17:10:09","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=753"},"modified":"2025-12-17T16:29:20","modified_gmt":"2025-12-17T16:29:20","slug":"numerical-integration-methods-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/numerical-integration-methods-apply-it\/","title":{"raw":"Numerical Integration Methods: Apply It","rendered":"Numerical Integration Methods: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Estimate definite integrals using the midpoint and trapezoidal rules<\/li>\r\n \t<li>Use Simpson's rule to find definite integrals with a specified accuracy<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Approximating Integrals in the Real World<\/h2>\r\nWhen exact antiderivatives aren\u2019t available, scientists and engineers use numerical integration to approximate values. For example, when modeling curved surfaces, predicting population changes, or estimating travel distances, approximation methods like the midpoint rule, trapezoidal rule, and simpson\u2019s rule give reliable results. Let\u2019s practice these techniques.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p data-start=\"861\" data-end=\"1057\">[ohm_question hide_question_numbers=1]313371[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p data-start=\"1557\" data-end=\"1658\">[ohm_question hide_question_numbers=1]313382[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p data-start=\"2198\" data-end=\"2393\">[ohm_question hide_question_numbers=1]313384[\/ohm_question]<\/p>\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p data-start=\"2860\" data-end=\"2976\">[ohm_question hide_question_numbers=1]313387[\/ohm_question]<\/p>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Estimate definite integrals using the midpoint and trapezoidal rules<\/li>\n<li>Use Simpson&#8217;s rule to find definite integrals with a specified accuracy<\/li>\n<\/ul>\n<\/section>\n<h2>Approximating Integrals in the Real World<\/h2>\n<p>When exact antiderivatives aren\u2019t available, scientists and engineers use numerical integration to approximate values. For example, when modeling curved surfaces, predicting population changes, or estimating travel distances, approximation methods like the midpoint rule, trapezoidal rule, and simpson\u2019s rule give reliable results. Let\u2019s practice these techniques.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p data-start=\"861\" data-end=\"1057\"><iframe loading=\"lazy\" id=\"ohm313371\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313371&theme=lumen&iframe_resize_id=ohm313371&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p data-start=\"1557\" data-end=\"1658\"><iframe loading=\"lazy\" id=\"ohm313382\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313382&theme=lumen&iframe_resize_id=ohm313382&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p data-start=\"2198\" data-end=\"2393\"><iframe loading=\"lazy\" id=\"ohm313384\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313384&theme=lumen&iframe_resize_id=ohm313384&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p data-start=\"2860\" data-end=\"2976\"><iframe loading=\"lazy\" id=\"ohm313387\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=313387&theme=lumen&iframe_resize_id=ohm313387&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":9,"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\/753"}],"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":14,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/753\/revisions"}],"predecessor-version":[{"id":2512,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/753\/revisions\/2512"}],"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\/753\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=753"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=753"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=753"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=753"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}