{"id":140,"date":"2021-07-30T17:19:58","date_gmt":"2021-07-30T17:19:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=140"},"modified":"2022-11-01T04:32:11","modified_gmt":"2022-11-01T04:32:11","slug":"summary-of-triple-integrals-in-cylindrical-and-spherical-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-triple-integrals-in-cylindrical-and-spherical-coordinates\/","title":{"raw":"Summary of Triple Integrals in Cylindrical and Spherical Coordinates","rendered":"Summary of Triple Integrals in Cylindrical and Spherical Coordinates"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>To evaluate a triple integral in cylindrical coordinates, use the iterated integral<\/li>\r\n \t<li>To evaluate a triple integral in spherical coordinates, use the iterated integral<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Triple integral in cylindrical coordinates\r\n<\/strong>[latex]\\underset{B}{\\displaystyle\\iiint} g(x,y,z)dV=\\underset{B}{\\displaystyle\\iiint} g(r\\cos\\theta,r\\sin\\theta,z)r dr d\\theta dz=\\underset{B}{\\displaystyle\\iiint} f(r,\\theta,z)r dr d\\theta dz=[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Triple integral in spherical coordinates<\/strong>\r\n[latex]\\underset{B}{\\displaystyle\\iiint} f(\\rho,\\theta,\\varphi)\\rho^{2}\\sin\\varphi{d}\\rho{d}\\varphi{d}\\theta=\\displaystyle\\int_{\\varphi=\\gamma}^{\\varphi=\\psi}\\displaystyle\\int_{\\theta=\\alpha}^{\\theta=\\beta}\\displaystyle\\int_{\\rho=a}^{\\rho=b} \\rho^{2}\\sin\\varphi{d}\\rho{d}\\varphi{d}\\theta[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>triple integral in cylindrical coordinates<\/dt>\r\n \t<dd>the limit of a triple Riemann sum, provided the following limit exists:[latex]{\\displaystyle\\lim_{l,m,n\\to\\infty}{\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}f({r^{*}}_{i,j,k}, {{\\theta}^{*}}_{i,j,k}, {{z}^{*}}_{i,j,k}){r^{*}}_{i,j,k}{\\Delta}r{\\Delta}{\\theta}{\\Delta}{z}}}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>triple integral in spherical coordinates<\/dt>\r\n \t<dd>the limit of a triple Riemann sum, provided the following limit exists:\u00a0[latex]{\\displaystyle\\lim_{l,m,n\\to\\infty}{\\displaystyle\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}f({{\\rho}^{*}}_{i,j,k}, {{\\theta}^{*}}_{i,j,k}, {{\\varphi}^{*}}_{i,j,k})({{\\rho}^{*}}_{i,j,k})^{2}\\sin{\\varphi}{\\Delta}{\\rho}{\\Delta}{\\theta}{\\Delta}{\\varphi}}}[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>To evaluate a triple integral in cylindrical coordinates, use the iterated integral<\/li>\n<li>To evaluate a triple integral in spherical coordinates, use the iterated integral<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Triple integral in cylindrical coordinates<br \/>\n<\/strong>[latex]\\underset{B}{\\displaystyle\\iiint} g(x,y,z)dV=\\underset{B}{\\displaystyle\\iiint} g(r\\cos\\theta,r\\sin\\theta,z)r dr d\\theta dz=\\underset{B}{\\displaystyle\\iiint} f(r,\\theta,z)r dr d\\theta dz=[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Triple integral in spherical coordinates<\/strong><br \/>\n[latex]\\underset{B}{\\displaystyle\\iiint} f(\\rho,\\theta,\\varphi)\\rho^{2}\\sin\\varphi{d}\\rho{d}\\varphi{d}\\theta=\\displaystyle\\int_{\\varphi=\\gamma}^{\\varphi=\\psi}\\displaystyle\\int_{\\theta=\\alpha}^{\\theta=\\beta}\\displaystyle\\int_{\\rho=a}^{\\rho=b} \\rho^{2}\\sin\\varphi{d}\\rho{d}\\varphi{d}\\theta[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>triple integral in cylindrical coordinates<\/dt>\n<dd>the limit of a triple Riemann sum, provided the following limit exists:[latex]{\\displaystyle\\lim_{l,m,n\\to\\infty}{\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}f({r^{*}}_{i,j,k}, {{\\theta}^{*}}_{i,j,k}, {{z}^{*}}_{i,j,k}){r^{*}}_{i,j,k}{\\Delta}r{\\Delta}{\\theta}{\\Delta}{z}}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>triple integral in spherical coordinates<\/dt>\n<dd>the limit of a triple Riemann sum, provided the following limit exists:\u00a0[latex]{\\displaystyle\\lim_{l,m,n\\to\\infty}{\\displaystyle\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}f({{\\rho}^{*}}_{i,j,k}, {{\\theta}^{*}}_{i,j,k}, {{\\varphi}^{*}}_{i,j,k})({{\\rho}^{*}}_{i,j,k})^{2}\\sin{\\varphi}{\\Delta}{\\rho}{\\Delta}{\\theta}{\\Delta}{\\varphi}}}[\/latex]<\/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-140\">\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 3. <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-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/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-3\/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 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