{"id":256,"date":"2022-02-23T13:23:25","date_gmt":"2022-02-23T18:23:25","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/multivariatecalculus\/?post_type=chapter&p=256"},"modified":"2022-03-03T22:57:28","modified_gmt":"2022-03-04T03:57:28","slug":"unit-12-line-integrals","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/chapter\/unit-12-line-integrals\/","title":{"raw":"Unit 12: Line Integrals","rendered":"Unit 12: Line Integrals"},"content":{"raw":"

The Concept<\/h2>\r\nFor a single-variable integral<\/strong> in 2D, [latex]\\int_a^b f(x)\\,dx[\/latex], we integrate function [latex]f(x)[\/latex] along [latex]x[\/latex] in 2D and it represents the area inbetween the curve, [latex]y=f(x)[\/latex], and a segment of [latex]x[\/latex]-axis from [latex]a[\/latex] to [latex]b[\/latex].\r\n\r\nA line integral<\/strong> in 3D shares a similar idea to a single-variable integral in 2D. A line integral<\/strong>, [latex]\\int_C f(x,y)\\,ds[\/latex], integrates the surface function, [latex]z=f(x,y)[\/latex], along a 2D curve segment [latex]C[\/latex] on the [latex]xy[\/latex]-plane, instead of [latex]x[\/latex] on the [latex]x[\/latex]-axis or [latex]y[\/latex] on the [latex]y[\/latex]-axis alone. This line segment, [latex]C[\/latex], is described by a vector function, [latex]r(t)=\\langle x(t), y(t) \\rangle[\/latex], where [latex]t=a[\/latex] and [latex]t=b[\/latex] map the start point and end point of [latex]C[\/latex], respectively. The differential element, [latex]ds[\/latex], represents the change of arc length of curve [latex]C[\/latex], i.e., [latex]ds=\\sqrt{(\\frac{dx}{dt})^2 + (\\frac{dy}{dt})^2} \\,\\,dt[\/latex]. Thus the line integral can be evaluated by the following single integral:\r\n

[latex]\\int_C f(x,y)\\,ds = \\int_a^b f(x(t),y(t))\\, \\sqrt{(\\frac{dx}{dt})^2 + (\\frac{dy}{dt})^2} \\,\\,dt[\/latex].<\/p>\r\nThe value of [latex]\\int_C f(x,y)\\,ds[\/latex] is the area of the \u201cwall\u201d, \"fence\" or \u201ccurtain\u201d whose base is the 2D curve [latex]C[\/latex] on the [latex]xy[\/latex]-plane and <\/span>and whose height is given by the function [latex]z=f(x,y)[\/latex].<\/span>\r\n\r\nThe concept of line integral can be extended to high dimensions. For example, [latex]\\int_C f(x,y,z)\\,ds [\/latex] integrates the function with three variables, [latex]w=f(x,y,z)[\/latex], along a 3D curve C that is parameterized by [latex]r(t) = \\langle x(t),y(t),z(t) \\rangle [\/latex]. It can be evaluated by a single integral as well, that is,<\/span><\/span>\r\n

[latex]\\int_C f(x,y,z)\\,ds = \\int_a^b f(x(t),y(t),z(t))\\, \\sqrt{(\\frac{dx}{dt})^2 + (\\frac{dy}{dt})^2+(\\frac{dz}{dt})^2} \\,\\,dt[\/latex].<\/p>\r\n\r\n

The Plot<\/h2>\r\nNow, you should engage with the 3D plot below to understand tangent planes [footnote]Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.[\/footnote]. Follow the steps below to apply changes to the plot and observe the effects:\r\n
    \r\n \t
  1. Input the function [latex]f(x, y)[\/latex].<\/li>\r\n \t
  2. Adjust the 2D curve [latex]C[\/latex] on the [latex]xy[\/latex]-plane.<\/li>\r\n \t
  3. Adjust the number of rectangular subareas, [latex]n[\/latex].<\/li>\r\n \t
  4. The estimation of the line integral is shown. The larger [latex]n[\/latex] is, the better the estimation is.<\/li>\r\n<\/ol>\r\n[h5p id=\"23\"]\r\n
    \r\n

    Self-Checking Questions<\/h2>\r\n<\/header>\r\n
    \r\n\r\nCheck your understanding by solving the following questions[footnote]Gilbert Strang, Edwin \u201cJed\u201d Herman,\u00a0 OpenStax,\u00a0Calculus Volume 3,\u00a0 Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).[\/footnote]<\/span>:\r\n
      \r\n \t
    1. Find the value of integral [latex]\\int_C(x^2+y^2)\\, ds[\/latex], where [latex]C[\/latex] is part of the helix parameterized by [latex]r(t)=\\langle cos t, sin t \\rangle, 0 \\leq t \\leq 2[\/latex].<\/li>\r\n \t
    2. Evaluate [latex]\\int_C \\frac{1}{x^2+y^2} \\, ds[\/latex] , over the line segment from [latex](1,1)[\/latex] to [latex](3,0)[\/latex].<\/li>\r\n<\/ol>\r\nUse the graph to find the answers to these questions.\r\n\r\n<\/div>\r\n<\/div>\r\n ","rendered":"

      The Concept<\/h2>\n

      For a single-variable integral<\/strong> in 2D, [latex]\\int_a^b f(x)\\,dx[\/latex], we integrate function [latex]f(x)[\/latex] along [latex]x[\/latex] in 2D and it represents the area inbetween the curve, [latex]y=f(x)[\/latex], and a segment of [latex]x[\/latex]-axis from [latex]a[\/latex] to [latex]b[\/latex].<\/p>\n

      A line integral<\/strong> in 3D shares a similar idea to a single-variable integral in 2D. A line integral<\/strong>, [latex]\\int_C f(x,y)\\,ds[\/latex], integrates the surface function, [latex]z=f(x,y)[\/latex], along a 2D curve segment [latex]C[\/latex] on the [latex]xy[\/latex]-plane, instead of [latex]x[\/latex] on the [latex]x[\/latex]-axis or [latex]y[\/latex] on the [latex]y[\/latex]-axis alone. This line segment, [latex]C[\/latex], is described by a vector function, [latex]r(t)=\\langle x(t), y(t) \\rangle[\/latex], where [latex]t=a[\/latex] and [latex]t=b[\/latex] map the start point and end point of [latex]C[\/latex], respectively. The differential element, [latex]ds[\/latex], represents the change of arc length of curve [latex]C[\/latex], i.e., [latex]ds=\\sqrt{(\\frac{dx}{dt})^2 + (\\frac{dy}{dt})^2} \\,\\,dt[\/latex]. Thus the line integral can be evaluated by the following single integral:<\/p>\n

      [latex]\\int_C f(x,y)\\,ds = \\int_a^b f(x(t),y(t))\\, \\sqrt{(\\frac{dx}{dt})^2 + (\\frac{dy}{dt})^2} \\,\\,dt[\/latex].<\/p>\n

      The value of [latex]\\int_C f(x,y)\\,ds[\/latex] is the area of the \u201cwall\u201d, “fence” or \u201ccurtain\u201d whose base is the 2D curve [latex]C[\/latex] on the [latex]xy[\/latex]-plane and <\/span>and whose height is given by the function [latex]z=f(x,y)[\/latex].<\/span><\/p>\n

      The concept of line integral can be extended to high dimensions. For example, [latex]\\int_C f(x,y,z)\\,ds[\/latex] integrates the function with three variables, [latex]w=f(x,y,z)[\/latex], along a 3D curve C that is parameterized by [latex]r(t) = \\langle x(t),y(t),z(t) \\rangle[\/latex]. It can be evaluated by a single integral as well, that is,<\/span><\/span><\/p>\n

      [latex]\\int_C f(x,y,z)\\,ds = \\int_a^b f(x(t),y(t),z(t))\\, \\sqrt{(\\frac{dx}{dt})^2 + (\\frac{dy}{dt})^2+(\\frac{dz}{dt})^2} \\,\\,dt[\/latex].<\/p>\n

      The Plot<\/h2>\n

      Now, you should engage with the 3D plot below to understand tangent planes [1]<\/sup><\/a>. Follow the steps below to apply changes to the plot and observe the effects:<\/p>\n

        \n
      1. Input the function [latex]f(x, y)[\/latex].<\/li>\n
      2. Adjust the 2D curve [latex]C[\/latex] on the [latex]xy[\/latex]-plane.<\/li>\n
      3. Adjust the number of rectangular subareas, [latex]n[\/latex].<\/li>\n
      4. The estimation of the line integral is shown. The larger [latex]n[\/latex] is, the better the estimation is.<\/li>\n<\/ol>\n
        \n
        <\/div>\n<\/div>\n
        \n
        \n

        Self-Checking Questions<\/h2>\n<\/header>\n
        \n

        Check your understanding by solving the following questions[2]<\/sup><\/a><\/span>:<\/p>\n

          \n
        1. Find the value of integral [latex]\\int_C(x^2+y^2)\\, ds[\/latex], where [latex]C[\/latex] is part of the helix parameterized by [latex]r(t)=\\langle cos t, sin t \\rangle, 0 \\leq t \\leq 2[\/latex].<\/li>\n
        2. Evaluate [latex]\\int_C \\frac{1}{x^2+y^2} \\, ds[\/latex] , over the line segment from [latex](1,1)[\/latex] to [latex](3,0)[\/latex].<\/li>\n<\/ol>\n

          Use the graph to find the answers to these questions.<\/p>\n<\/div>\n<\/div>\n

           <\/p>\n

          1. Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0. ↵<\/a><\/li>
          2. Gilbert Strang, Edwin \u201cJed\u201d Herman,\u00a0 OpenStax,\u00a0Calculus Volume 3,\u00a0 Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA). ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":391,"menu_order":12,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-256","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/256","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/users\/391"}],"version-history":[{"count":27,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/256\/revisions"}],"predecessor-version":[{"id":554,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/256\/revisions\/554"}],"part":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/256\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/media?parent=256"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=256"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/contributor?post=256"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/license?post=256"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}