{"id":695,"date":"2025-06-20T17:04:43","date_gmt":"2025-06-20T17:04:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=695"},"modified":"2025-09-08T15:41:29","modified_gmt":"2025-09-08T15:41:29","slug":"trigonometric-integrals-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/trigonometric-integrals-apply-it\/","title":{"raw":"Integration by Parts: Apply It","rendered":"Integration by Parts: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Recognize when to use integration by parts compared to other integration methods<\/li>\r\n \t<li>Use the integration by parts formula to solve indefinite integrals<\/li>\r\n \t<li>Apply integration by parts to evaluate definite integrals<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">From Lab to Life: Integration by Parts in Practice<\/h2>\r\nWelcome to a day in the life of an engineering consultant! You're working on various projects that require integration by parts to model real-world phenomena. From analyzing mechanical vibrations to processing electrical signals and calculating economic projections, each task demonstrates how integration by parts helps solve practical engineering and scientific problems.\r\n<h3>Damped Vibration Analysis<\/h3>\r\nA mechanical engineer is analyzing the displacement of a damped spring system. The velocity of the oscillating mass is given by [latex]v(t) = te^{-0.5t}[\/latex] meters per second, where [latex]t[\/latex] is time in seconds. To find the total displacement from [latex]t = 0[\/latex] to [latex]t = 4[\/latex] seconds, the engineer needs to evaluate:\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int_0^4 te^{-0.5t} \\, dt[\/latex]<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]310323[\/ohm_question]<\/section>\r\n<h3>Signal Processing Application<\/h3>\r\nAn electrical engineer is analyzing a communication signal with amplitude modulation. The power dissipated in a circuit component is proportional to:\r\n<p style=\"text-align: center;\">[latex]P(t) = \\int x\\cos(2x) \\, dx[\/latex]<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311156[\/ohm_question]<\/section>\r\n<h3>Economic Growth Model<\/h3>\r\nAn economist is modeling the present value of a continuously growing income stream. The present value of income that grows linearly with time and is continuously discounted is:\r\n<p style=\"text-align: center;\">[latex]PV = \\displaystyle\\int_0^{10} 5t e^{-0.1t} \\, dt[\/latex]<\/p>\r\nwhere [latex]t[\/latex] is time in years and all values are in thousands of dollars.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311221[\/ohm_question]<\/section>\r\n<h3>Heat Transfer Analysis<\/h3>\r\nA thermal engineer is analyzing heat dissipation in a cooling fin. The rate of heat transfer along the fin is described by:\r\n<p style=\"text-align: center;\">[latex]Q = k\\displaystyle\\int_0^L x^2 e^{-\\beta x} \\, dx[\/latex]<\/p>\r\nwhere [latex]L = 2[\/latex], [latex]\\beta = 1[\/latex], and [latex]k[\/latex] is a constant. To find the total heat transfer, evaluate:\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int_0^2 x^2 e^{-x} \\, dx[\/latex]<\/p>\r\n<p style=\"text-align: left;\">This integral requires applying integration by parts twice.<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311222[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Recognize when to use integration by parts compared to other integration methods<\/li>\n<li>Use the integration by parts formula to solve indefinite integrals<\/li>\n<li>Apply integration by parts to evaluate definite integrals<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">From Lab to Life: Integration by Parts in Practice<\/h2>\n<p>Welcome to a day in the life of an engineering consultant! You&#8217;re working on various projects that require integration by parts to model real-world phenomena. From analyzing mechanical vibrations to processing electrical signals and calculating economic projections, each task demonstrates how integration by parts helps solve practical engineering and scientific problems.<\/p>\n<h3>Damped Vibration Analysis<\/h3>\n<p>A mechanical engineer is analyzing the displacement of a damped spring system. The velocity of the oscillating mass is given by [latex]v(t) = te^{-0.5t}[\/latex] meters per second, where [latex]t[\/latex] is time in seconds. To find the total displacement from [latex]t = 0[\/latex] to [latex]t = 4[\/latex] seconds, the engineer needs to evaluate:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int_0^4 te^{-0.5t} \\, dt[\/latex]<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm310323\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=310323&theme=lumen&iframe_resize_id=ohm310323&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Signal Processing Application<\/h3>\n<p>An electrical engineer is analyzing a communication signal with amplitude modulation. The power dissipated in a circuit component is proportional to:<\/p>\n<p style=\"text-align: center;\">[latex]P(t) = \\int x\\cos(2x) \\, dx[\/latex]<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311156\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311156&theme=lumen&iframe_resize_id=ohm311156&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Economic Growth Model<\/h3>\n<p>An economist is modeling the present value of a continuously growing income stream. The present value of income that grows linearly with time and is continuously discounted is:<\/p>\n<p style=\"text-align: center;\">[latex]PV = \\displaystyle\\int_0^{10} 5t e^{-0.1t} \\, dt[\/latex]<\/p>\n<p>where [latex]t[\/latex] is time in years and all values are in thousands of dollars.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311221\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311221&theme=lumen&iframe_resize_id=ohm311221&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Heat Transfer Analysis<\/h3>\n<p>A thermal engineer is analyzing heat dissipation in a cooling fin. The rate of heat transfer along the fin is described by:<\/p>\n<p style=\"text-align: center;\">[latex]Q = k\\displaystyle\\int_0^L x^2 e^{-\\beta x} \\, dx[\/latex]<\/p>\n<p>where [latex]L = 2[\/latex], [latex]\\beta = 1[\/latex], and [latex]k[\/latex] is a constant. To find the total heat transfer, evaluate:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\int_0^2 x^2 e^{-x} \\, dx[\/latex]<\/p>\n<p style=\"text-align: left;\">This integral requires applying integration by parts twice.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311222\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311222&theme=lumen&iframe_resize_id=ohm311222&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":541,"module-header":"apply_it","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\/695"}],"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":15,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/695\/revisions"}],"predecessor-version":[{"id":2241,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/695\/revisions\/2241"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/541"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/695\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=695"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=695"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=695"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=695"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}