{"id":705,"date":"2025-06-20T17:06:24","date_gmt":"2025-06-20T17:06:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=705"},"modified":"2025-09-08T17:31:18","modified_gmt":"2025-09-08T17:31:18","slug":"trigonometric-integrals-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/trigonometric-integrals-apply-it-2\/","title":{"raw":"Trigonometric Integrals: Apply It","rendered":"Trigonometric Integrals: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Integrate expressions containing products and powers of sine and cosine<\/li>\r\n \t<li>Integrate expressions containing products and powers of tangent and secant<\/li>\r\n \t<li>Use reduction formulas to simplify and solve trigonometric integrals<\/li>\r\n \t<li>Integrate expressions containing square roots of sums or differences of squares<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Waves, Light, and Motion: Applications of Trigonometric Integrals<\/h2>\r\nFrom analyzing sound waves in music production to calculating electromagnetic fields in engineering, trigonometric integrals appear throughout science and technology. Today, you'll apply your integration skills to solve real-world problems involving periodic phenomena and geometric calculations. Let's explore how these mathematical techniques help us understand waves, optics, and mechanical systems.\r\n<h3>Sound Wave Energy<\/h3>\r\nAn audio engineer is analyzing the energy in a sound wave. The instantaneous power of a sound wave over one period is given by [latex]P(t) = \\sin^2(t)\\cos^2(t)[\/latex] watts, where [latex]t[\/latex] is measured in milliseconds from [latex]0[\/latex] to [latex]\\pi[\/latex] ms.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311224[\/ohm_question]<\/section>\r\n<h3>Optical Path Length<\/h3>\r\nA lens designer needs to calculate the optical path length through a curved lens surface. The path length through the lens at position [latex]x[\/latex] (in centimeters) is given by:\r\n<p style=\"text-align: center;\">[latex]L = \\int_{-2}^{2} \\sqrt{16-x^2} \\, dx[\/latex]<\/p>\r\nThis represents the cross-sectional area of a semicircular lens with radius [latex]4[\/latex] cm.\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311225[\/ohm_question]<\/section>\r\n<h3>Mechanical Vibration Analysis<\/h3>\r\nAn engineer is studying a vibrating beam whose displacement follows a complex pattern. The work done by the restoring force over one complete cycle involves evaluating:\r\n<p style=\"text-align: center;\">[latex]W = \\int_0^{\\pi\/2} \\tan^3(x)\\sec(x) \\, dx[\/latex]<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311227[\/ohm_question]<\/section>\r\n<h3>Electromagnetic Field Calculation<\/h3>\r\nIn designing an antenna, an engineer needs to evaluate the electric field intensity pattern given by:\r\n<p style=\"text-align: center;\">[latex]E = \\int_0^{\\pi\/4} \\sec^3(\\theta) \\, d\\theta[\/latex]<\/p>\r\n\r\n<section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]311228[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Integrate expressions containing products and powers of sine and cosine<\/li>\n<li>Integrate expressions containing products and powers of tangent and secant<\/li>\n<li>Use reduction formulas to simplify and solve trigonometric integrals<\/li>\n<li>Integrate expressions containing square roots of sums or differences of squares<\/li>\n<\/ul>\n<\/section>\n<h2>Waves, Light, and Motion: Applications of Trigonometric Integrals<\/h2>\n<p>From analyzing sound waves in music production to calculating electromagnetic fields in engineering, trigonometric integrals appear throughout science and technology. Today, you&#8217;ll apply your integration skills to solve real-world problems involving periodic phenomena and geometric calculations. Let&#8217;s explore how these mathematical techniques help us understand waves, optics, and mechanical systems.<\/p>\n<h3>Sound Wave Energy<\/h3>\n<p>An audio engineer is analyzing the energy in a sound wave. The instantaneous power of a sound wave over one period is given by [latex]P(t) = \\sin^2(t)\\cos^2(t)[\/latex] watts, where [latex]t[\/latex] is measured in milliseconds from [latex]0[\/latex] to [latex]\\pi[\/latex] ms.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311224\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311224&theme=lumen&iframe_resize_id=ohm311224&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Optical Path Length<\/h3>\n<p>A lens designer needs to calculate the optical path length through a curved lens surface. The path length through the lens at position [latex]x[\/latex] (in centimeters) is given by:<\/p>\n<p style=\"text-align: center;\">[latex]L = \\int_{-2}^{2} \\sqrt{16-x^2} \\, dx[\/latex]<\/p>\n<p>This represents the cross-sectional area of a semicircular lens with radius [latex]4[\/latex] cm.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311225\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311225&theme=lumen&iframe_resize_id=ohm311225&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Mechanical Vibration Analysis<\/h3>\n<p>An engineer is studying a vibrating beam whose displacement follows a complex pattern. The work done by the restoring force over one complete cycle involves evaluating:<\/p>\n<p style=\"text-align: center;\">[latex]W = \\int_0^{\\pi\/2} \\tan^3(x)\\sec(x) \\, dx[\/latex]<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311227\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311227&theme=lumen&iframe_resize_id=ohm311227&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Electromagnetic Field Calculation<\/h3>\n<p>In designing an antenna, an engineer needs to evaluate the electric field intensity pattern given by:<\/p>\n<p style=\"text-align: center;\">[latex]E = \\int_0^{\\pi\/4} \\sec^3(\\theta) \\, d\\theta[\/latex]<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm311228\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=311228&theme=lumen&iframe_resize_id=ohm311228&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":18,"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":"- 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\/705"}],"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":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/705\/revisions"}],"predecessor-version":[{"id":2244,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/705\/revisions\/2244"}],"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\/705\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=705"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=705"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=705"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=705"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}