{"id":134,"date":"2021-07-30T17:19:16","date_gmt":"2021-07-30T17:19:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=134"},"modified":"2022-11-01T04:20:22","modified_gmt":"2022-11-01T04:20:22","slug":"summary-of-double-integrals-over-general-regions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-double-integrals-over-general-regions\/","title":{"raw":"Summary of Double Integrals over General Regions","rendered":"Summary of Double Integrals over General Regions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>A general bounded region [latex]D[\/latex]\u00a0on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.<\/li>\r\n \t<li>To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.<\/li>\r\n \t<li>We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.<\/li>\r\n \t<li>We can use Fubini\u2019s theorem for improper integrals to evaluate some types of improper integrals.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Iterated integral over a Type I region\r\n<\/strong>[latex]\\displaystyle\\iint_{D} f(x,y)dA=\\displaystyle\\iint_{D} f(x,y)dydx=\\displaystyle\\int_{a}^{b}\\left[\\displaystyle\\int_{g_{1}(x)}^{g_{2}(x)}f(x,y)dy\\right] dx[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Iterated integral over a Type II region<\/strong>\r\n[latex]\\displaystyle\\iint_{D} f(x,y)dA=\\displaystyle\\iint_{D} f(x,y)dydx=\\displaystyle\\int_{c}^{d}\\left[\\displaystyle\\int_{h_{1}(y)}^{h_{2}(y)}f(x,y)dx\\right] dy[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>improper double integral<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a double integral over an unbounded region or of an unbounded function<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>Type I<\/dt>\r\n \t<dd>a region\u00a0[latex]\\bf{D}[\/latex] in the [latex]xy[\/latex]-plane is Type I\u00a0if it lies between two vertical lines and the graphs of two continuous functions [latex]g_{1}(x)[\/latex] and\u00a0[latex]g_{2}(x)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>Type II<\/dt>\r\n \t<dd>a region [latex]\\bf{D}[\/latex]<strong>\u00a0<\/strong>in the [latex]xy[\/latex]-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions [latex]h_{1}(y)[\/latex]\u00a0and\u00a0[latex]h_{2}(y)[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>A general bounded region [latex]D[\/latex]\u00a0on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.<\/li>\n<li>To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.<\/li>\n<li>We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.<\/li>\n<li>We can use Fubini\u2019s theorem for improper integrals to evaluate some types of improper integrals.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Iterated integral over a Type I region<br \/>\n<\/strong>[latex]\\displaystyle\\iint_{D} f(x,y)dA=\\displaystyle\\iint_{D} f(x,y)dydx=\\displaystyle\\int_{a}^{b}\\left[\\displaystyle\\int_{g_{1}(x)}^{g_{2}(x)}f(x,y)dy\\right] dx[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Iterated integral over a Type II region<\/strong><br \/>\n[latex]\\displaystyle\\iint_{D} f(x,y)dA=\\displaystyle\\iint_{D} f(x,y)dydx=\\displaystyle\\int_{c}^{d}\\left[\\displaystyle\\int_{h_{1}(y)}^{h_{2}(y)}f(x,y)dx\\right] dy[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>improper double integral<\/dt>\n<dd><span style=\"font-size: 1em;\">a double integral over an unbounded region or of an unbounded function<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Type I<\/dt>\n<dd>a region\u00a0[latex]\\bf{D}[\/latex] in the [latex]xy[\/latex]-plane is Type I\u00a0if it lies between two vertical lines and the graphs of two continuous functions [latex]g_{1}(x)[\/latex] and\u00a0[latex]g_{2}(x)[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Type II<\/dt>\n<dd>a region [latex]\\bf{D}[\/latex]<strong>\u00a0<\/strong>in the [latex]xy[\/latex]-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions [latex]h_{1}(y)[\/latex]\u00a0and\u00a0[latex]h_{2}(y)[\/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-134\">\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":11,"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-134","chapter","type-chapter","status-publish","hentry"],"part":23,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/134","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":12,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/134\/revisions"}],"predecessor-version":[{"id":3775,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/134\/revisions\/3775"}],"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\/134\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=134"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=134"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=134"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}