{"id":390,"date":"2025-02-13T19:44:32","date_gmt":"2025-02-13T19:44:32","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/the-fundamental-theorem-of-calculus-apply-it\/"},"modified":"2025-02-13T19:44:32","modified_gmt":"2025-02-13T19:44:32","slug":"the-fundamental-theorem-of-calculus-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/the-fundamental-theorem-of-calculus-apply-it\/","title":{"raw":"The Fundamental Theorem of Calculus: Apply It","rendered":"The Fundamental Theorem of Calculus: Apply It"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Understand the Mean Value Theorem for Integrals and both components of the Fundamental Theorem of Calculus<\/li>\n\t<li>Use the Fundamental Theorem of Calculus to find derivatives of integral functions and calculate definite integrals<\/li>\n\t<li>Describe how differentiation and integration are interconnected<\/li>\n<\/ul>\n<\/section>\n<h2>A Parachutist in Free Fall<\/h2>\n<p>&nbsp;<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"701\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204117\/CNX_Calc_Figure_05_03_006.jpg\" alt=\"Two skydivers free falling in the sky.\" width=\"701\" height=\"462\"> Figure 5. Skydivers can adjust the velocity of their dive by changing the position of their body during the free fall. (credit: Jeremy T. Lock)[\/caption]\n\n<p id=\"fs-id1170572218426\">Julie is an avid <span class=\"no-emphasis\">skydiver<\/span>. She has more than [latex]300[\/latex] jumps under her belt and has mastered the art of making adjustments to her body position in the air to control how fast she falls. If she arches her back and points her belly toward the ground, she reaches a terminal velocity of approximately [latex]120[\/latex] mph ([latex]176[\/latex] ft\/sec). If, instead, she orients her body with her head straight down, she falls faster, reaching a terminal velocity of [latex]150[\/latex] mph ([latex]220[\/latex] ft\/sec).<\/p>\n<p id=\"fs-id1170572218437\">Since Julie will be moving (falling) in a downward direction, we assume the downward direction is positive to simplify our calculations. Julie executes her jumps from an altitude of [latex]12,500[\/latex] ft. After she exits the aircraft, she immediately starts falling at a velocity given by [latex]v(t)=32t.[\/latex] She continues to accelerate according to this velocity function until she reaches terminal velocity. After she reaches terminal velocity, her speed remains constant until she pulls her ripcord and slows down to land.<\/p>\n<p id=\"fs-id1170572419103\">On her first jump of the day, Julie orients herself in the slower \u201cbelly down\u201d position (terminal velocity is [latex]176[\/latex] ft\/sec). Using this information, answer the following questions.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p>[ohm_question hide_question_numbers=1]288189[\/ohm_question]<\/p>\n<\/section>\n<p>We can now set up an expression that represents the distance Julie falls after [latex]30[\/latex] sec.<\/p>\n<p style=\"text-align: center;\">[latex]\\int_0^{5.5} 32t \\, dt + \\int_{5.5}^{30} 176 \\, dt[\/latex]<br>\n<br>\n<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p>[ohm_question hide_question_numbers=1]288190[\/ohm_question]<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p>[ohm_question hide_question_numbers=1]288191[\/ohm_question]<\/p>\n<\/section>\n<p>On Julie\u2019s second jump of the day, she decides she wants to fall a little faster and orients herself in the \u201chead down\u201d position. Her terminal velocity in this position is [latex]220[\/latex] ft\/sec. Answer these questions based on this velocity:&nbsp;<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p>[ohm_question hide_question_numbers=1]288192[\/ohm_question]<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p>[ohm_question hide_question_numbers=1]288193[\/ohm_question]<\/p>\n<\/section>\n<p>Some jumpers wear \u201c<span class=\"no-emphasis\">wingsuits<\/span>\u201d (see Figure 6). These suits have fabric panels between the arms and legs and allow the wearer to glide around in a free fall, much like a flying squirrel. (Indeed, the suits are sometimes called \u201cflying squirrel suits.\u201d) When wearing these suits, terminal velocity can be reduced to about [latex]30[\/latex] mph ([latex]44[\/latex] ft\/sec), allowing the wearers a much longer time in the air. Wingsuit flyers still use parachutes to land; although the vertical velocities are within the margin of safety, horizontal velocities can exceed [latex]70[\/latex] mph, much too fast to land safely.<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"700\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204124\/CNX_Calc_Figure_05_03_007.jpg\" alt=\"A person falling in a wingsuit, which works to reduce the vertical velocity of a skydiver\u2019s fall.\" width=\"700\" height=\"392\"> Figure 6. The fabric panels on the arms and legs of a wingsuit work to reduce the vertical velocity of a skydiver\u2019s fall. (credit: Richard Schneider)[\/caption]\n\n<p id=\"fs-id1163722683509\">Answer the following question based on the velocity in a wingsuit.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p>[ohm_question hide_question_numbers=1]288194[\/ohm_question]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand the Mean Value Theorem for Integrals and both components of the Fundamental Theorem of Calculus<\/li>\n<li>Use the Fundamental Theorem of Calculus to find derivatives of integral functions and calculate definite integrals<\/li>\n<li>Describe how differentiation and integration are interconnected<\/li>\n<\/ul>\n<\/section>\n<h2>A Parachutist in Free Fall<\/h2>\n<p>&nbsp;<\/p>\n<figure style=\"width: 701px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204117\/CNX_Calc_Figure_05_03_006.jpg\" alt=\"Two skydivers free falling in the sky.\" width=\"701\" height=\"462\" \/><figcaption class=\"wp-caption-text\">Figure 5. Skydivers can adjust the velocity of their dive by changing the position of their body during the free fall. (credit: Jeremy T. Lock)<\/figcaption><\/figure>\n<p id=\"fs-id1170572218426\">Julie is an avid <span class=\"no-emphasis\">skydiver<\/span>. She has more than [latex]300[\/latex] jumps under her belt and has mastered the art of making adjustments to her body position in the air to control how fast she falls. If she arches her back and points her belly toward the ground, she reaches a terminal velocity of approximately [latex]120[\/latex] mph ([latex]176[\/latex] ft\/sec). If, instead, she orients her body with her head straight down, she falls faster, reaching a terminal velocity of [latex]150[\/latex] mph ([latex]220[\/latex] ft\/sec).<\/p>\n<p id=\"fs-id1170572218437\">Since Julie will be moving (falling) in a downward direction, we assume the downward direction is positive to simplify our calculations. Julie executes her jumps from an altitude of [latex]12,500[\/latex] ft. After she exits the aircraft, she immediately starts falling at a velocity given by [latex]v(t)=32t.[\/latex] She continues to accelerate according to this velocity function until she reaches terminal velocity. After she reaches terminal velocity, her speed remains constant until she pulls her ripcord and slows down to land.<\/p>\n<p id=\"fs-id1170572419103\">On her first jump of the day, Julie orients herself in the slower \u201cbelly down\u201d position (terminal velocity is [latex]176[\/latex] ft\/sec). Using this information, answer the following questions.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm288189\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288189&theme=lumen&iframe_resize_id=ohm288189&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>We can now set up an expression that represents the distance Julie falls after [latex]30[\/latex] sec.<\/p>\n<p style=\"text-align: center;\">[latex]\\int_0^{5.5} 32t \\, dt + \\int_{5.5}^{30} 176 \\, dt[\/latex]<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm288190\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288190&theme=lumen&iframe_resize_id=ohm288190&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm288191\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288191&theme=lumen&iframe_resize_id=ohm288191&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>On Julie\u2019s second jump of the day, she decides she wants to fall a little faster and orients herself in the \u201chead down\u201d position. Her terminal velocity in this position is [latex]220[\/latex] ft\/sec. Answer these questions based on this velocity:&nbsp;<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm288192\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288192&theme=lumen&iframe_resize_id=ohm288192&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm288193\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288193&theme=lumen&iframe_resize_id=ohm288193&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<p>Some jumpers wear \u201c<span class=\"no-emphasis\">wingsuits<\/span>\u201d (see Figure 6). These suits have fabric panels between the arms and legs and allow the wearer to glide around in a free fall, much like a flying squirrel. (Indeed, the suits are sometimes called \u201cflying squirrel suits.\u201d) When wearing these suits, terminal velocity can be reduced to about [latex]30[\/latex] mph ([latex]44[\/latex] ft\/sec), allowing the wearers a much longer time in the air. Wingsuit flyers still use parachutes to land; although the vertical velocities are within the margin of safety, horizontal velocities can exceed [latex]70[\/latex] mph, much too fast to land safely.<\/p>\n<figure style=\"width: 700px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11204124\/CNX_Calc_Figure_05_03_007.jpg\" alt=\"A person falling in a wingsuit, which works to reduce the vertical velocity of a skydiver\u2019s fall.\" width=\"700\" height=\"392\" \/><figcaption class=\"wp-caption-text\">Figure 6. The fabric panels on the arms and legs of a wingsuit work to reduce the vertical velocity of a skydiver\u2019s fall. (credit: Richard Schneider)<\/figcaption><\/figure>\n<p id=\"fs-id1163722683509\">Answer the following question based on the velocity in a wingsuit.<\/p>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<iframe loading=\"lazy\" id=\"ohm288194\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288194&theme=lumen&iframe_resize_id=ohm288194&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":6,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":371,"module-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\/390"}],"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\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/390\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/371"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/390\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=390"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=390"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=390"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=390"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}