{"id":236,"date":"2022-02-23T12:25:07","date_gmt":"2022-02-23T17:25:07","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/multivariatecalculus\/?post_type=chapter&p=236"},"modified":"2024-01-30T22:52:53","modified_gmt":"2024-01-31T03:52:53","slug":"unit-9-triple-integrals-in-cylindrical-coordinates","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/chapter\/unit-9-triple-integrals-in-cylindrical-coordinates\/","title":{"raw":"Unit 9: Triple Integrals in Cylindrical Coordinates","rendered":"Unit 9: Triple Integrals in Cylindrical Coordinates"},"content":{"raw":"

The Concept<\/h2>\r\nCylindrical coordinates<\/strong> are a simple extension of 2D polar coordinates<\/strong> to 3D. Recall that, in Unit 7, the position of a point in 2D (i.e., [latex]xy[\/latex]-plane) can be described using polar coordinates [latex](r, \\theta)[\/latex], where [latex]r[\/latex] is the distance of the point from the origin and [latex]\\theta[\/latex] is the angle between the [latex]x[\/latex]-axis and the line segment from the origin to the point. With the addition of a third dimension, [latex]z[\/latex]-axis from the Cartesian (i.e., rectangular) coordinate system, we are able to describe a point in 3D cylindrical coordinates, i.e., [latex](r, \\theta, z)[\/latex].\r\n\r\nCylindrical coordinates simply combine the polar coordinates in the [latex]xy[\/latex]-plane with the usual [latex]z[\/latex] coordinate of Cartesian coordinates. To form the cylindrical coordinates of a point [latex]P[\/latex], simply project it down to a point [latex]Q[\/latex] in the [latex]xy[\/latex]-plane. Then, take the polar coordinates [latex](r, \\theta)[\/latex] of the point [latex]Q[\/latex]. The third cylindrical coordinate is the same as the usual [latex]z[\/latex]-coordinate. It is the signed distance of point [latex]P[\/latex] to the [latex]xy[\/latex]-plane.\r\n

The Plot<\/h2>\r\nNow, you should engage with the 3D plot below to understand triple integrals in cylindrical coordinates[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. Change the bounds on the triple integral in cylindrical coordinates where and represent the outermost bounds. [latex]\\alpha[\/latex] and [latex]\\beta[\/latex] are constants and they are the lower and upper bounds of angle [latex]\\theta[\/latex].\u00a0 [latex]r_1[\/latex] and [latex]r_2[\/latex] are functions of [latex]\\theta[\/latex], and they are the lower and upper bounds of [latex]r[\/latex].\u00a0 [latex]u_1[\/latex] and [latex]u_2[\/latex] are functions of [latex]r[\/latex] and [latex]\\theta[\/latex], and they are the lower and upper bounds of [latex]z[\/latex].<\/li>\r\n \t
  2. You may also change the grid size of the 3D solid depicted on the screen for the function [latex]f(x,y,z)[\/latex].<\/li>\r\n<\/ol>\r\n[h5p id=\"21\"]\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\nEvaluate the triple integrals [latex]f(x,y,z)[\/latex] over the solid [latex]E[\/latex].\r\n
      \r\n \t
    1. [latex]E = \\{(x,y,z)| x^2+y^2 \\leq 9, x \\geq 0, y\\geq 0, 0 \\leq z \\leq 1\\}, f(x,y,z) = z[\/latex]<\/li>\r\n \t
    2. [latex]E = \\{(x,y,z)| 1 \\leq x^2+y^2 \\leq 9,\u00a0 y \\geq 0, 0 \\leq z \\leq 1\\}, f(x,y,z) = x^2+y^2[\/latex]<\/li>\r\n<\/ol>\r\nUse the graph to find the answers to these questions.\r\n\r\n<\/div>\r\n<\/div>","rendered":"

      The Concept<\/h2>\n

      Cylindrical coordinates<\/strong> are a simple extension of 2D polar coordinates<\/strong> to 3D. Recall that, in Unit 7, the position of a point in 2D (i.e., [latex]xy[\/latex]-plane) can be described using polar coordinates [latex](r, \\theta)[\/latex], where [latex]r[\/latex] is the distance of the point from the origin and [latex]\\theta[\/latex] is the angle between the [latex]x[\/latex]-axis and the line segment from the origin to the point. With the addition of a third dimension, [latex]z[\/latex]-axis from the Cartesian (i.e., rectangular) coordinate system, we are able to describe a point in 3D cylindrical coordinates, i.e., [latex](r, \\theta, z)[\/latex].<\/p>\n

      Cylindrical coordinates simply combine the polar coordinates in the [latex]xy[\/latex]-plane with the usual [latex]z[\/latex] coordinate of Cartesian coordinates. To form the cylindrical coordinates of a point [latex]P[\/latex], simply project it down to a point [latex]Q[\/latex] in the [latex]xy[\/latex]-plane. Then, take the polar coordinates [latex](r, \\theta)[\/latex] of the point [latex]Q[\/latex]. The third cylindrical coordinate is the same as the usual [latex]z[\/latex]-coordinate. It is the signed distance of point [latex]P[\/latex] to the [latex]xy[\/latex]-plane.<\/p>\n

      The Plot<\/h2>\n

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

        \n
      1. Change the bounds on the triple integral in cylindrical coordinates where and represent the outermost bounds. [latex]\\alpha[\/latex] and [latex]\\beta[\/latex] are constants and they are the lower and upper bounds of angle [latex]\\theta[\/latex].\u00a0 [latex]r_1[\/latex] and [latex]r_2[\/latex] are functions of [latex]\\theta[\/latex], and they are the lower and upper bounds of [latex]r[\/latex].\u00a0 [latex]u_1[\/latex] and [latex]u_2[\/latex] are functions of [latex]r[\/latex] and [latex]\\theta[\/latex], and they are the lower and upper bounds of [latex]z[\/latex].<\/li>\n
      2. You may also change the grid size of the 3D solid depicted on the screen for the function [latex]f(x,y,z)[\/latex].<\/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

        Evaluate the triple integrals [latex]f(x,y,z)[\/latex] over the solid [latex]E[\/latex].<\/p>\n

          \n
        1. [latex]E = \\{(x,y,z)| x^2+y^2 \\leq 9, x \\geq 0, y\\geq 0, 0 \\leq z \\leq 1\\}, f(x,y,z) = z[\/latex]<\/li>\n
        2. [latex]E = \\{(x,y,z)| 1 \\leq x^2+y^2 \\leq 9,\u00a0 y \\geq 0, 0 \\leq z \\leq 1\\}, f(x,y,z) = x^2+y^2[\/latex]<\/li>\n<\/ol>\n

          Use the graph to find the answers to these questions.<\/p>\n<\/div>\n<\/div>\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":9,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-236","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/236","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":18,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/236\/revisions"}],"predecessor-version":[{"id":643,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/236\/revisions\/643"}],"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\/236\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/media?parent=236"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=236"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/contributor?post=236"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/license?post=236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}