{"id":136,"date":"2021-07-30T17:19:32","date_gmt":"2021-07-30T17:19:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=136"},"modified":"2022-11-01T04:24:41","modified_gmt":"2022-11-01T04:24:41","slug":"summary-of-double-integrals-in-polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-double-integrals-in-polar-coordinates\/","title":{"raw":"Summary of Double Integrals in Polar Coordinates","rendered":"Summary of Double Integrals in Polar Coordinates"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.<\/li>\r\n \t<li>The area [latex]dA[\/latex]\u00a0in polar coordinates becomes [latex]rdrd\\theta[\/latex].<\/li>\r\n \t<li>Use [latex]x=r\\cos\\theta[\/latex], [latex]y=r\\sin\\theta[\/latex], and [latex]dA=rdrd\\theta[\/latex]\u00a0to convert an integral in rectangular coordinates to an integral in polar coordinates.<\/li>\r\n \t<li>Use [latex]r^{2}=x^{2}+y^{2}[\/latex] and [latex]\\theta=\\tan^{-1}\\left(\\frac{y}{x}\\right)[\/latex] to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.<\/li>\r\n \t<li>To find the volume in polar coordinates bounded above by a surface [latex]z=f(r,\\theta)[\/latex] over a region on the [latex]xy[\/latex]-plane, use a double integral in polar coordinates.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Double integral over a polar rectangular region\r\n<\/strong>[latex]\\underset{R}{\\displaystyle\\iint} f(r,\\theta)dA = \\underset{m,n\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f({r_{ij}}^{\\ast},{\\theta_{ij}}^{\\ast})\\Delta{A} =\\underset{m,n\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f({r_{ij}}^{\\ast},{\\theta_{ij}}^{\\ast}){r_{ij}}^{\\ast}\\Delta{r}\\Delta{\\theta}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Double integral over a general polar region<\/strong><strong>\r\n<\/strong>[latex]\\underset{D}{\\displaystyle\\iint} f(r,\\theta)r dr d\\theta=\\displaystyle\\int_{\\theta=\\alpha}^{\\theta=\\beta} \\displaystyle\\int_{r=h_{1}(\\theta)}^{r=h_{2}(\\theta)}f(r,\\theta)r dr d\\theta[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>polar rectangle<\/dt>\r\n \t<dd>the region enclosed between the circles [latex]r=a[\/latex] and [latex]r=b[\/latex]\u00a0and the angles [latex]\\theta = \\alpha[\/latex]\u00a0and [latex]\\theta = \\beta[\/latex]; it is described as\u00a0[latex]{\\bf{R}}=\\{(r,{\\theta}) | a{\\leq}r{\\leq}b, {\\alpha}{\\leq}{\\theta}{\\leq}{\\beta}\\}[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.<\/li>\n<li>The area [latex]dA[\/latex]\u00a0in polar coordinates becomes [latex]rdrd\\theta[\/latex].<\/li>\n<li>Use [latex]x=r\\cos\\theta[\/latex], [latex]y=r\\sin\\theta[\/latex], and [latex]dA=rdrd\\theta[\/latex]\u00a0to convert an integral in rectangular coordinates to an integral in polar coordinates.<\/li>\n<li>Use [latex]r^{2}=x^{2}+y^{2}[\/latex] and [latex]\\theta=\\tan^{-1}\\left(\\frac{y}{x}\\right)[\/latex] to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.<\/li>\n<li>To find the volume in polar coordinates bounded above by a surface [latex]z=f(r,\\theta)[\/latex] over a region on the [latex]xy[\/latex]-plane, use a double integral in polar coordinates.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Double integral over a polar rectangular region<br \/>\n<\/strong>[latex]\\underset{R}{\\displaystyle\\iint} f(r,\\theta)dA = \\underset{m,n\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f({r_{ij}}^{\\ast},{\\theta_{ij}}^{\\ast})\\Delta{A} =\\underset{m,n\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f({r_{ij}}^{\\ast},{\\theta_{ij}}^{\\ast}){r_{ij}}^{\\ast}\\Delta{r}\\Delta{\\theta}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Double integral over a general polar region<\/strong><strong><br \/>\n<\/strong>[latex]\\underset{D}{\\displaystyle\\iint} f(r,\\theta)r dr d\\theta=\\displaystyle\\int_{\\theta=\\alpha}^{\\theta=\\beta} \\displaystyle\\int_{r=h_{1}(\\theta)}^{r=h_{2}(\\theta)}f(r,\\theta)r dr d\\theta[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>polar rectangle<\/dt>\n<dd>the region enclosed between the circles [latex]r=a[\/latex] and [latex]r=b[\/latex]\u00a0and the angles [latex]\\theta = \\alpha[\/latex]\u00a0and [latex]\\theta = \\beta[\/latex]; it is described as\u00a0[latex]{\\bf{R}}=\\{(r,{\\theta}) | a{\\leq}r{\\leq}b, {\\alpha}{\\leq}{\\theta}{\\leq}{\\beta}\\}[\/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-136\">\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 <\/section>","protected":false},"author":17533,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/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-136","chapter","type-chapter","status-publish","hentry"],"part":23,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/136","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":13,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/136\/revisions"}],"predecessor-version":[{"id":3776,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/136\/revisions\/3776"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/23"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/136\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=136"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=136"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=136"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}