{"id":716,"date":"2021-05-10T19:10:42","date_gmt":"2021-05-10T19:10:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=716"},"modified":"2021-11-17T23:47:46","modified_gmt":"2021-11-17T23:47:46","slug":"summary-of-calculus-of-parametric-curves","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-calculus-of-parametric-curves\/","title":{"raw":"Summary of Calculus of Parametric Curves","rendered":"Summary of Calculus of Parametric Curves"},"content":{"raw":"<section id=\"fs-id1167794065921\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1167794065929\" data-bullet-style=\"bullet\">\r\n \t<li>The derivative of the parametrically defined curve [latex]x=x\\left(t\\right)[\/latex] and [latex]y=y\\left(t\\right)[\/latex] can be calculated using the formula [latex]\\frac{dy}{dx}=\\frac{{y}^{\\prime }\\left(t\\right)}{{x}^{\\prime }\\left(t\\right)}[\/latex]. Using the derivative, we can find the equation of a tangent line to a parametric curve.<\/li>\r\n \t<li>The area between a parametric curve and the <em data-effect=\"italics\">x<\/em>-axis can be determined by using the formula [latex]A={\\displaystyle\\int }_{{t}_{1}}^{{t}_{2}}y\\left(t\\right){x}^{\\prime }\\left(t\\right)dt[\/latex].<\/li>\r\n \t<li>The arc length of a parametric curve can be calculated by using the formula [latex]s={\\displaystyle\\int }_{{t}_{1}}^{{t}_{2}}\\sqrt{{\\left(\\frac{dx}{dt}\\right)}^{2}+{\\left(\\frac{dy}{dt}\\right)}^{2}}dt[\/latex].<\/li>\r\n \t<li>The surface area of a volume of revolution revolved around the <em data-effect=\"italics\">x<\/em>-axis is given by [latex]S=2\\pi {\\displaystyle\\int }_{a}^{b}y\\left(t\\right)\\sqrt{{\\left({x}^{\\prime }\\left(t\\right)\\right)}^{2}+{\\left({y}^{\\prime }\\left(t\\right)\\right)}^{2}}dt[\/latex]. If the curve is revolved around the <em data-effect=\"italics\">y<\/em>-axis, then the formula is [latex]S=2\\pi {\\displaystyle\\int }_{a}^{b}x\\left(t\\right)\\sqrt{{\\left({x}^{\\prime }\\left(t\\right)\\right)}^{2}+{\\left({y}^{\\prime }\\left(t\\right)\\right)}^{2}}dt[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1167794056011\" class=\"key-equations\" data-depth=\"1\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1167794056018\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Derivative of parametric equations<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\frac{dy}{dx}=\\frac{\\frac{dy}{dt}}{\\frac{dx}{dt}}=\\frac{{y}^{\\prime }\\left(t\\right)}{{x}^{\\prime }\\left(t\\right)}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Second-order derivative of parametric equations<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]\\frac{{d}^{2}y}{d{x}^{2}}=\\frac{d}{dx}\\left(\\frac{dy}{dx}\\right)=\\frac{\\left(\\frac{d}{dt}\\right)\\left(\\frac{dy}{dx}\\right)}{\\frac{dx}{dt}}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Area under a parametric curve<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]A={\\displaystyle\\int }_{a}^{b}y\\left(t\\right){x}^{\\prime }\\left(t\\right)dt[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Arc length of a parametric curve<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]s={\\displaystyle\\int }_{{t}_{1}}^{{t}_{2}}\\sqrt{{\\left(\\frac{dx}{dt}\\right)}^{2}+{\\left(\\frac{dy}{dt}\\right)}^{2}}dt[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Surface area generated by a parametric curve<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]S=2\\pi {\\displaystyle\\int }_{a}^{b}y\\left(t\\right)\\sqrt{{\\left({x}^{\\prime }\\left(t\\right)\\right)}^{2}+{\\left({y}^{\\prime }\\left(t\\right)\\right)}^{2}}dt[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>","rendered":"<section id=\"fs-id1167794065921\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1167794065929\" data-bullet-style=\"bullet\">\n<li>The derivative of the parametrically defined curve [latex]x=x\\left(t\\right)[\/latex] and [latex]y=y\\left(t\\right)[\/latex] can be calculated using the formula [latex]\\frac{dy}{dx}=\\frac{{y}^{\\prime }\\left(t\\right)}{{x}^{\\prime }\\left(t\\right)}[\/latex]. Using the derivative, we can find the equation of a tangent line to a parametric curve.<\/li>\n<li>The area between a parametric curve and the <em data-effect=\"italics\">x<\/em>-axis can be determined by using the formula [latex]A={\\displaystyle\\int }_{{t}_{1}}^{{t}_{2}}y\\left(t\\right){x}^{\\prime }\\left(t\\right)dt[\/latex].<\/li>\n<li>The arc length of a parametric curve can be calculated by using the formula [latex]s={\\displaystyle\\int }_{{t}_{1}}^{{t}_{2}}\\sqrt{{\\left(\\frac{dx}{dt}\\right)}^{2}+{\\left(\\frac{dy}{dt}\\right)}^{2}}dt[\/latex].<\/li>\n<li>The surface area of a volume of revolution revolved around the <em data-effect=\"italics\">x<\/em>-axis is given by [latex]S=2\\pi {\\displaystyle\\int }_{a}^{b}y\\left(t\\right)\\sqrt{{\\left({x}^{\\prime }\\left(t\\right)\\right)}^{2}+{\\left({y}^{\\prime }\\left(t\\right)\\right)}^{2}}dt[\/latex]. If the curve is revolved around the <em data-effect=\"italics\">y<\/em>-axis, then the formula is [latex]S=2\\pi {\\displaystyle\\int }_{a}^{b}x\\left(t\\right)\\sqrt{{\\left({x}^{\\prime }\\left(t\\right)\\right)}^{2}+{\\left({y}^{\\prime }\\left(t\\right)\\right)}^{2}}dt[\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1167794056011\" class=\"key-equations\" data-depth=\"1\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1167794056018\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Derivative of parametric equations<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\frac{dy}{dx}=\\frac{\\frac{dy}{dt}}{\\frac{dx}{dt}}=\\frac{{y}^{\\prime }\\left(t\\right)}{{x}^{\\prime }\\left(t\\right)}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Second-order derivative of parametric equations<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]\\frac{{d}^{2}y}{d{x}^{2}}=\\frac{d}{dx}\\left(\\frac{dy}{dx}\\right)=\\frac{\\left(\\frac{d}{dt}\\right)\\left(\\frac{dy}{dx}\\right)}{\\frac{dx}{dt}}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Area under a parametric curve<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]A={\\displaystyle\\int }_{a}^{b}y\\left(t\\right){x}^{\\prime }\\left(t\\right)dt[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Arc length of a parametric curve<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]s={\\displaystyle\\int }_{{t}_{1}}^{{t}_{2}}\\sqrt{{\\left(\\frac{dx}{dt}\\right)}^{2}+{\\left(\\frac{dy}{dt}\\right)}^{2}}dt[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Surface area generated by a parametric curve<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]S=2\\pi {\\displaystyle\\int }_{a}^{b}y\\left(t\\right)\\sqrt{{\\left({x}^{\\prime }\\left(t\\right)\\right)}^{2}+{\\left({y}^{\\prime }\\left(t\\right)\\right)}^{2}}dt[\/latex]<\/li>\n<\/ul>\n<\/section>\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-716\">\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":416434,"menu_order":9,"template":"","meta":{"_candela_citation":"{\"1\":{\"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-716","chapter","type-chapter","status-publish","hentry"],"part":162,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/716","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\/416434"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/716\/revisions"}],"predecessor-version":[{"id":1074,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/716\/revisions\/1074"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/162"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/716\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=716"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=716"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=716"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=716"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}