{"id":167,"date":"2022-02-22T22:25:35","date_gmt":"2022-02-23T03:25:35","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/multivariatecalculus\/?post_type=chapter&#038;p=167"},"modified":"2022-03-03T22:27:21","modified_gmt":"2022-03-04T03:27:21","slug":"unit-6-double-integrals-over-the-general-region","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/chapter\/unit-6-double-integrals-over-the-general-region\/","title":{"raw":"Unit 6: Double Integrals Over the General Region","rendered":"Unit 6: Double Integrals Over the General Region"},"content":{"raw":"<h2>The Concept<\/h2>\r\nNow, you should engage with the 3D plot below to understand the double integral over the general region (i.e., non-rectangular region). There are two types of double integrals.\r\n<ul>\r\n \t<li><strong>Type I double integral<\/strong>: [latex]\\int_{a}^{b} \\int_{h(x)}^{g(x)} f(x,y)\\,dy dx[\/latex], where [latex]x=a[\/latex] and [latex]x=b[\/latex] are the lower and upper bounds of [latex]x[\/latex]; [latex]y=h(x)[\/latex] and [latex]y=g(x)[\/latex] are the lower and upper bounds of [latex]y[\/latex].<\/li>\r\n \t<li><strong>Type II double integral: <\/strong>[latex]\\int_{a}^{b} \\int_{h(x)}^{g(x)} f(x,y)\\,dx dy[\/latex], where [latex]y=a[\/latex] and [latex]y=b[\/latex] are the lower and upper bounds of [latex]y[\/latex]; [latex]x=h(y)[\/latex] and [latex]x=g(y)[\/latex]are the lower and upper bounds of [latex]y[\/latex].<\/li>\r\n<\/ul>\r\nYou may notice that the bounds of outer integrals ([latex]a[\/latex] and [latex]b[\/latex]) for both Type I and Type II integrals are constants; these two integrals are \u201csymmetric\u201d \u2013 if you switch [latex]x[\/latex] and [latex]y[\/latex] in Type I, you get Type II and vice versa.\r\n<h2>The Plot<\/h2>\r\nNow, you should engage with the plot below to understand double integrals with general regions[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<ol>\r\n \t<li style=\"font-weight: 400\">Assume you have a Type I integral [latex]\\int_{0}^{1} \\int_{-x}^{x^2} y^2 x\\,dy dx[\/latex]. Input [latex]y^2 x[\/latex] into the [latex]f(x,y)[\/latex] input function section.<\/li>\r\n \t<li style=\"font-weight: 400\">Input [latex]\\textrm{If} (0 \\leq x \\leq 1, x^2)[\/latex] into the Upper [latex]y[\/latex] function, i.e., [latex]g(x)[\/latex] section.<\/li>\r\n \t<li style=\"font-weight: 400\">Input [latex]\\textrm{If} (0 \\leq x \\leq 1, -x)[\/latex] into the lower [latex]y[\/latex] function, i.e., [latex]h(x)[\/latex] section.<\/li>\r\n \t<li style=\"font-weight: 400\">Use the slider for the value of x to see the change of the area of the cross-section, [latex]A(x)[\/latex].<\/li>\r\n \t<li style=\"font-weight: 400\">The result of this double integral is dynamically calculated at the bottom.<\/li>\r\n \t<li style=\"font-weight: 400\">You can also use this plot for the Type II integral by switching [latex]x[\/latex] and [latex]y.[\/latex]<\/li>\r\n<\/ol>\r\n<span>[h5p id=\"30\"]<\/span>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<h2 class=\"textbox__title\">Self-Checking Questions<\/h2>\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nCheck your understanding by solving the following questions<span>[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<ol>\r\n \t<li>[latex]\\int_{0}^{1} \\int_{2\\sqrt{x}}^{2\\sqrt{x}+1} xy+1\\,dy dx[\/latex].<\/li>\r\n \t<li>[latex] \\int_{0}^{1} \\int_{-\\sqrt{1-y^2}}^{\\sqrt{1-y^2}} 2x+4x^3\\,dx dy[\/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":"<h2>The Concept<\/h2>\n<p>Now, you should engage with the 3D plot below to understand the double integral over the general region (i.e., non-rectangular region). There are two types of double integrals.<\/p>\n<ul>\n<li><strong>Type I double integral<\/strong>: [latex]\\int_{a}^{b} \\int_{h(x)}^{g(x)} f(x,y)\\,dy dx[\/latex], where [latex]x=a[\/latex] and [latex]x=b[\/latex] are the lower and upper bounds of [latex]x[\/latex]; [latex]y=h(x)[\/latex] and [latex]y=g(x)[\/latex] are the lower and upper bounds of [latex]y[\/latex].<\/li>\n<li><strong>Type II double integral: <\/strong>[latex]\\int_{a}^{b} \\int_{h(x)}^{g(x)} f(x,y)\\,dx dy[\/latex], where [latex]y=a[\/latex] and [latex]y=b[\/latex] are the lower and upper bounds of [latex]y[\/latex]; [latex]x=h(y)[\/latex] and [latex]x=g(y)[\/latex]are the lower and upper bounds of [latex]y[\/latex].<\/li>\n<\/ul>\n<p>You may notice that the bounds of outer integrals ([latex]a[\/latex] and [latex]b[\/latex]) for both Type I and Type II integrals are constants; these two integrals are \u201csymmetric\u201d \u2013 if you switch [latex]x[\/latex] and [latex]y[\/latex] in Type I, you get Type II and vice versa.<\/p>\n<h2>The Plot<\/h2>\n<p>Now, you should engage with the plot below to understand double integrals with general regions<a class=\"footnote\" title=\"Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.\" id=\"return-footnote-167-1\" href=\"#footnote-167-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>. Follow the steps below to apply changes to the plot and observe the effects:<\/p>\n<ol>\n<li style=\"font-weight: 400\">Assume you have a Type I integral [latex]\\int_{0}^{1} \\int_{-x}^{x^2} y^2 x\\,dy dx[\/latex]. Input [latex]y^2 x[\/latex] into the [latex]f(x,y)[\/latex] input function section.<\/li>\n<li style=\"font-weight: 400\">Input [latex]\\textrm{If} (0 \\leq x \\leq 1, x^2)[\/latex] into the Upper [latex]y[\/latex] function, i.e., [latex]g(x)[\/latex] section.<\/li>\n<li style=\"font-weight: 400\">Input [latex]\\textrm{If} (0 \\leq x \\leq 1, -x)[\/latex] into the lower [latex]y[\/latex] function, i.e., [latex]h(x)[\/latex] section.<\/li>\n<li style=\"font-weight: 400\">Use the slider for the value of x to see the change of the area of the cross-section, [latex]A(x)[\/latex].<\/li>\n<li style=\"font-weight: 400\">The result of this double integral is dynamically calculated at the bottom.<\/li>\n<li style=\"font-weight: 400\">You can also use this plot for the Type II integral by switching [latex]x[\/latex] and [latex]y.[\/latex]<\/li>\n<\/ol>\n<p><span><\/p>\n<div id=\"h5p-30\">\n<div class=\"h5p-content\" data-content-id=\"30\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<h2 class=\"textbox__title\">Self-Checking Questions<\/h2>\n<\/header>\n<div class=\"textbox__content\">\n<p>Check your understanding by solving the following questions<span><a class=\"footnote\" title=\"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).\" id=\"return-footnote-167-2\" href=\"#footnote-167-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/span>:<\/p>\n<ol>\n<li>[latex]\\int_{0}^{1} \\int_{2\\sqrt{x}}^{2\\sqrt{x}+1} xy+1\\,dy dx[\/latex].<\/li>\n<li>[latex]\\int_{0}^{1} \\int_{-\\sqrt{1-y^2}}^{\\sqrt{1-y^2}} 2x+4x^3\\,dx dy[\/latex]<\/li>\n<\/ol>\n<p>Use the graph to find the answers to these questions.<\/p>\n<\/div>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-167-1\">Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0. <a href=\"#return-footnote-167-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-167-2\">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 href=\"#return-footnote-167-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":391,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-167","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/167","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":24,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/167\/revisions"}],"predecessor-version":[{"id":547,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapters\/167\/revisions\/547"}],"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\/167\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/media?parent=167"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=167"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/contributor?post=167"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/wp-json\/wp\/v2\/license?post=167"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}