{"id":1177,"date":"2021-06-30T17:02:05","date_gmt":"2021-06-30T17:02:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-arc-length-of-a-curve-and-surface-area\/"},"modified":"2021-11-17T02:06:31","modified_gmt":"2021-11-17T02:06:31","slug":"summary-of-arc-length-of-a-curve-and-surface-area","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-arc-length-of-a-curve-and-surface-area\/","title":{"raw":"Summary of Arc Length of a Curve and Surface Area","rendered":"Summary of Arc Length of a Curve and Surface Area"},"content":{"raw":"<div id=\"fs-id1167793477144\" class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167793505417\">\r\n \t<li>The arc length of a curve can be calculated using a definite integral.<\/li>\r\n \t<li>The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions of [latex]y.[\/latex]<\/li>\r\n \t<li>The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution.<\/li>\r\n \t<li>The integrals generated by both the arc length and surface area formulas are often difficult to evaluate. It may be necessary to use a computer or calculator to approximate the values of the integrals.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1167794058940\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1167793404628\">\r\n \t<li><strong>Arc Length of a Function of [latex]x[\/latex]<\/strong>\r\n[latex]\\text{Arc Length}={\\displaystyle\\int }_{a}^{b}\\sqrt{1+{\\left[{f}^{\\prime }(x)\\right]}^{2}}dx[\/latex]<\/li>\r\n \t<li><strong>Arc Length of a Function of [latex]y[\/latex]<\/strong>\r\n[latex]\\text{Arc Length}={\\displaystyle\\int }_{c}^{d}\\sqrt{1+{\\left[{g}^{\\prime }(y)\\right]}^{2}}dy[\/latex]<\/li>\r\n \t<li><strong>Surface Area of a Function of [latex]x[\/latex]<\/strong>\r\n[latex]\\text{Surface Area}={\\displaystyle\\int }_{a}^{b}(2\\pi f(x)\\sqrt{1+{({f}^{\\prime }(x))}^{2}})dx[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1167793720071\" class=\"definition\">\r\n \t<dt>arc length<\/dt>\r\n \t<dd id=\"fs-id1167793720076\">the arc length of a curve can be thought of as the distance a person would travel along the path of the curve<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167793455097\" class=\"definition\">\r\n \t<dt>frustum<\/dt>\r\n \t<dd id=\"fs-id1167793455103\">a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167793455108\" class=\"definition\">\r\n \t<dt>surface area<\/dt>\r\n \t<dd id=\"fs-id1167793455113\">the surface area of a solid is the total area of the outer layer of the object; for objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces<\/dd>\r\n<\/dl>","rendered":"<div id=\"fs-id1167793477144\" class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167793505417\">\n<li>The arc length of a curve can be calculated using a definite integral.<\/li>\n<li>The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions of [latex]y.[\/latex]<\/li>\n<li>The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution.<\/li>\n<li>The integrals generated by both the arc length and surface area formulas are often difficult to evaluate. It may be necessary to use a computer or calculator to approximate the values of the integrals.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1167794058940\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1167793404628\">\n<li><strong>Arc Length of a Function of [latex]x[\/latex]<\/strong><br \/>\n[latex]\\text{Arc Length}={\\displaystyle\\int }_{a}^{b}\\sqrt{1+{\\left[{f}^{\\prime }(x)\\right]}^{2}}dx[\/latex]<\/li>\n<li><strong>Arc Length of a Function of [latex]y[\/latex]<\/strong><br \/>\n[latex]\\text{Arc Length}={\\displaystyle\\int }_{c}^{d}\\sqrt{1+{\\left[{g}^{\\prime }(y)\\right]}^{2}}dy[\/latex]<\/li>\n<li><strong>Surface Area of a Function of [latex]x[\/latex]<\/strong><br \/>\n[latex]\\text{Surface Area}={\\displaystyle\\int }_{a}^{b}(2\\pi f(x)\\sqrt{1+{({f}^{\\prime }(x))}^{2}})dx[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167793720071\" class=\"definition\">\n<dt>arc length<\/dt>\n<dd id=\"fs-id1167793720076\">the arc length of a curve can be thought of as the distance a person would travel along the path of the curve<\/dd>\n<\/dl>\n<dl id=\"fs-id1167793455097\" class=\"definition\">\n<dt>frustum<\/dt>\n<dd id=\"fs-id1167793455103\">a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base<\/dd>\n<\/dl>\n<dl id=\"fs-id1167793455108\" class=\"definition\">\n<dt>surface area<\/dt>\n<dd id=\"fs-id1167793455113\">the surface area of a solid is the total area of the outer layer of the object; for objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1177\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 2. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":17,"template":"","meta":{"_candela_citation":"{\"2\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"}}","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1177","chapter","type-chapter","status-publish","hentry"],"part":1160,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1177","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1177\/revisions"}],"predecessor-version":[{"id":2508,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1177\/revisions\/2508"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/1160"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1177\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1177"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1177"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1177"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1177"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}