{"id":752,"date":"2025-10-09T15:36:45","date_gmt":"2025-10-09T15:36:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=752"},"modified":"2025-11-05T18:04:36","modified_gmt":"2025-11-05T18:04:36","slug":"line-integrals-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/line-integrals-learn-it-1\/","title":{"raw":"Line Integrals: Learn It 1","rendered":"Line Integrals: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Calculate scalar line integrals<\/li>\r\n \t<li>Calculate vector line integrals<\/li>\r\n \t<li>Apply line integrals to compute work<\/li>\r\n \t<li>Describe flux and circulation<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Line Integrals<\/h2>\r\n<p class=\"whitespace-normal break-words\">Single-variable integrals of the form [latex]\\int_a^b f(x) dx[\/latex] integrate over an interval [latex][a,b][\/latex] on the [latex]x[\/latex]-axis. We can think of this interval as a curve\u2014specifically, a line segment with endpoints [latex](a,0)[\/latex] and [latex](b,0)[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">But what if we want to integrate over any curve in the plane or in space, not just a line segment on the [latex]x[\/latex]-axis? This requires a new kind of integral called a <strong>line integral<\/strong>.<\/p>\r\n\r\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">Line integrals have many applications to engineering and physics. They allow us to make several useful generalizations of the Fundamental Theorem of Calculus, and they are closely connected to the properties of vector fields.<\/section>\r\n<p class=\"whitespace-normal break-words\">There are two types of line integrals:<\/p>\r\n\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Scalar line integrals<\/strong> integrate a scalar function over a curve in a plane or in space.<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>Vector line integrals<\/strong> integrate a vector field over a curve in a plane or in space.<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">We'll begin by studying scalar line integrals.<\/p>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Calculate scalar line integrals<\/li>\n<li>Calculate vector line integrals<\/li>\n<li>Apply line integrals to compute work<\/li>\n<li>Describe flux and circulation<\/li>\n<\/ul>\n<\/section>\n<h2>Line Integrals<\/h2>\n<p class=\"whitespace-normal break-words\">Single-variable integrals of the form [latex]\\int_a^b f(x) dx[\/latex] integrate over an interval [latex][a,b][\/latex] on the [latex]x[\/latex]-axis. We can think of this interval as a curve\u2014specifically, a line segment with endpoints [latex](a,0)[\/latex] and [latex](b,0)[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">But what if we want to integrate over any curve in the plane or in space, not just a line segment on the [latex]x[\/latex]-axis? This requires a new kind of integral called a <strong>line integral<\/strong>.<\/p>\n<section class=\"textbox connectIt\" aria-label=\"Connect It\">Line integrals have many applications to engineering and physics. They allow us to make several useful generalizations of the Fundamental Theorem of Calculus, and they are closely connected to the properties of vector fields.<\/section>\n<p class=\"whitespace-normal break-words\">There are two types of line integrals:<\/p>\n<ul>\n<li class=\"whitespace-normal break-words\"><strong>Scalar line integrals<\/strong> integrate a scalar function over a curve in a plane or in space.<\/li>\n<li class=\"whitespace-normal break-words\"><strong>Vector line integrals<\/strong> integrate a vector field over a curve in a plane or in space.<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">We&#8217;ll begin by studying scalar line integrals.<\/p>\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":[],"class_list":["post-752","chapter","type-chapter","status-publish","hentry"],"part":436,"module-header":"learn_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\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/752","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/752\/revisions"}],"predecessor-version":[{"id":2008,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/752\/revisions\/2008"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/436"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/752\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=752"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=752"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=752"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=752"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}