{"id":973,"date":"2025-06-20T17:26:05","date_gmt":"2025-06-20T17:26:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=973"},"modified":"2025-07-30T14:03:10","modified_gmt":"2025-07-30T14:03:10","slug":"parametric-curves-and-their-applications-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/parametric-curves-and-their-applications-cheat-sheet\/","title":{"raw":"Parametric Curves and Their Applications: Cheat Sheet","rendered":"Parametric Curves and Their Applications: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Fundamentals of Parametric Equations<\/strong>\r\n<ul id=\"fs-id1169293333646\" data-bullet-style=\"bullet\">\r\n \t<li>Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity.<\/li>\r\n \t<li>It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve.<\/li>\r\n \t<li>There is always more than one way to parameterize a curve.<\/li>\r\n \t<li>Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates.<\/li>\r\n<\/ul>\r\n<strong>Calculus with Parametric Curves<\/strong>\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<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<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169293394905\">\r\n \t<dt>cusp<\/dt>\r\n \t<dd id=\"fs-id1169293394910\">a pointed end or part where two curves meet<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293394894\">\r\n \t<dt>cycloid<\/dt>\r\n \t<dd id=\"fs-id1169293394900\">the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293394914\">\r\n \t<dt>orientation<\/dt>\r\n \t<dd id=\"fs-id1169293394920\">the direction that a point moves on a graph as the parameter increases<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293394924\">\r\n \t<dt>parameter<\/dt>\r\n \t<dd id=\"fs-id1169293394929\">an independent variable that both <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> depend on in a parametric curve; usually represented by the variable <em data-effect=\"italics\">t<\/em><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293394948\">\r\n \t<dt>parametric curve<\/dt>\r\n \t<dd id=\"fs-id1169293394954\">the graph of the parametric equations [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] over an interval [latex]a\\le t\\le b[\/latex] combined with the equations<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293296425\">\r\n \t<dt>parametric equations<\/dt>\r\n \t<dd id=\"fs-id1169293296430\">the equations [latex]x=x\\left(t\\right)[\/latex] and [latex]y=y\\left(t\\right)[\/latex] that define a parametric curve<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293296469\">\r\n \t<dt>parameterization of a curve<\/dt>\r\n \t<dd id=\"fs-id1169293296474\">rewriting the equation of a curve defined by a function [latex]y=f\\left(x\\right)[\/latex] as parametric equations<\/dd>\r\n<\/dl>","rendered":"<h2>Essential Concepts<\/h2>\n<p><strong>Fundamentals of Parametric Equations<\/strong><\/p>\n<ul id=\"fs-id1169293333646\" data-bullet-style=\"bullet\">\n<li>Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity.<\/li>\n<li>It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve.<\/li>\n<li>There is always more than one way to parameterize a curve.<\/li>\n<li>Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates.<\/li>\n<\/ul>\n<p><strong>Calculus with Parametric Curves<\/strong><\/p>\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<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<h2>Glossary<\/h2>\n<dl id=\"fs-id1169293394905\">\n<dt>cusp<\/dt>\n<dd id=\"fs-id1169293394910\">a pointed end or part where two curves meet<\/dd>\n<\/dl>\n<dl id=\"fs-id1169293394894\">\n<dt>cycloid<\/dt>\n<dd id=\"fs-id1169293394900\">the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage<\/dd>\n<\/dl>\n<dl id=\"fs-id1169293394914\">\n<dt>orientation<\/dt>\n<dd id=\"fs-id1169293394920\">the direction that a point moves on a graph as the parameter increases<\/dd>\n<\/dl>\n<dl id=\"fs-id1169293394924\">\n<dt>parameter<\/dt>\n<dd id=\"fs-id1169293394929\">an independent variable that both <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> depend on in a parametric curve; usually represented by the variable <em data-effect=\"italics\">t<\/em><\/dd>\n<\/dl>\n<dl id=\"fs-id1169293394948\">\n<dt>parametric curve<\/dt>\n<dd id=\"fs-id1169293394954\">the graph of the parametric equations [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] over an interval [latex]a\\le t\\le b[\/latex] combined with the equations<\/dd>\n<\/dl>\n<dl id=\"fs-id1169293296425\">\n<dt>parametric equations<\/dt>\n<dd id=\"fs-id1169293296430\">the equations [latex]x=x\\left(t\\right)[\/latex] and [latex]y=y\\left(t\\right)[\/latex] that define a parametric curve<\/dd>\n<\/dl>\n<dl id=\"fs-id1169293296469\">\n<dt>parameterization of a curve<\/dt>\n<dd id=\"fs-id1169293296474\">rewriting the equation of a curve defined by a function [latex]y=f\\left(x\\right)[\/latex] as parametric equations<\/dd>\n<\/dl>\n","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":675,"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\/973"}],"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":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/973\/revisions"}],"predecessor-version":[{"id":1681,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/973\/revisions\/1681"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/675"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/973\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=973"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=973"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=973"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=973"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}