{"id":1191,"date":"2025-10-15T15:16:40","date_gmt":"2025-10-15T15:16:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=1191"},"modified":"2025-10-29T15:35:41","modified_gmt":"2025-10-29T15:35:41","slug":"double-integrals-over-basic-regions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/double-integrals-over-basic-regions-learn-it-4\/","title":{"raw":"Double Integrals over Basic Regions: Learn It 4","rendered":"Double Integrals over Basic Regions: Learn It 4"},"content":{"raw":"<h2 data-type=\"title\">General Regions of Integration<\/h2>\r\n<p class=\"whitespace-normal break-words\">So far we have computed double integrals over rectangular regions. However, most regions we encounter in applications are not rectangular.<\/p>\r\nConsider a general bounded region [latex]D[\/latex] on a plane as shown in Figure 1. Since [latex]D[\/latex] is bounded, there must exist a rectangular region [latex]R[\/latex] that encloses [latex]D[\/latex]. In other words, we can find a rectangle [latex]R[\/latex] such that [latex]D[\/latex] is a subset of [latex]R[\/latex], written as [latex]D \\subseteq R[\/latex].\r\n\r\n[caption id=\"attachment_1317\" align=\"aligncenter\" width=\"417\"]<img class=\"size-full wp-image-1317\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/23152039\/5-2-1.jpeg\" alt=\"A rectangle R with a shape D inside of it. Inside D, there is a point labeled g(x, y) = f(x, y). Outside D but still inside R, there is a point labeled g(x, y) = 0.\" width=\"417\" height=\"310\" \/> Figure 1. For a region [latex]\\small{D}[\/latex] that is a subset of\u00a0[latex]\\small{R}[\/latex], we can define a function\u00a0[latex]\\small{g(x,y)}[\/latex] to equal\u00a0[latex]\\small{f(x,y)}[\/latex] at every point in\u00a0[latex]\\small{D}[\/latex] and\u00a0[latex]\\small{0}[\/latex] at every point of\u00a0[latex]\\small{R}[\/latex] not in\u00a0[latex]\\small{D}[\/latex].[\/caption]\r\n<p class=\"whitespace-normal break-words\">Now suppose [latex]z=f(x, y)[\/latex] is defined on this general bounded region [latex]D[\/latex]. To develop double integrals of [latex]f[\/latex] over [latex]D[\/latex], we extend the definition of the function to include all points on a rectangular region [latex]R[\/latex] that contains [latex]D[\/latex]. We define a new function [latex]g(x, y)[\/latex] on [latex]R[\/latex] as follows:<\/p>\r\n<p style=\"text-align: center;\">[latex] g(x,y)= \\begin{cases} f(x,y) &amp; \\text{if } (x,y) \\text{ is in } D \\\\ 0 &amp; \\text{if } (x,y) \\text{ is in } R \\text{ but not in } D \\end{cases} [\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">This extension allows us to use the integration techniques from rectangular regions. The function [latex]g(x,y)[\/latex] equals [latex]f(x,y)[\/latex] inside [latex]D[\/latex] and equals zero outside [latex]D[\/latex] but within [latex]R[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">We assume the boundary of [latex]D[\/latex] is a piecewise smooth and continuous simple closed curve to avoid technical difficulties. Under this assumption, [latex]g(x, y)[\/latex] is integrable over the rectangular region [latex]R[\/latex] as long as [latex]f(x,y)[\/latex] is integrable over [latex]D[\/latex]. For now we focus on describing different types of regions rather than the technical details of the function.<\/p>\r\n<p class=\"whitespace-normal break-words\">We consider two types of planar bounded regions.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>type I and type II regions<\/h3>\r\n<p id=\"fs-id1167793884044\">A region [latex]D[\/latex] in the [latex](x, y)[\/latex]-plane is of\u00a0<strong><span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term209\" data-type=\"term\">Type I<\/span>\u00a0<\/strong>if it lies between two vertical lines and the graphs of two continuous functions [latex]g_1(x)[\/latex] and [latex]g_2(x)[\/latex]. That is,,<\/p>\r\n<p style=\"text-align: center;\">[latex]{D} = \\left \\{(x,y) \\mid a \\leq x \\leq \\ b,g_1 \\leq y \\leq g_2 \\ (x) \\right \\}[\/latex].<\/p>\r\n<p id=\"fs-id1167794337439\">A region [latex]D[\/latex] in the [latex]xy[\/latex] plane is of\u00a0<strong><span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term210\" data-type=\"term\">Type II<\/span><\/strong> if it lies between two horizontal lines and the graphs of two continuous functions [latex]h_1(y)[\/latex] and [latex]h_2(y)[\/latex]. That is,<\/p>\r\n<p style=\"text-align: center;\">[latex]{D} = \\left \\{(x,y) \\mid c \\leq y \\leq \\ d,h_1 \\leq x \\leq h_2 \\ (y) \\right \\}[\/latex].<\/p>\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">Figures 2 and 3 illustrate the geometric interpretation of Type I and Type II regions.<\/p>\r\n\r\n[caption id=\"attachment_1320\" align=\"aligncenter\" width=\"963\"]<img class=\"size-full wp-image-1320\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/23155027\/5-2-2.jpeg\" alt=\"The graphs showing a region marked D. In all instances, between a and b, there is a shape that is defined by two functions g1(x) and g2(x). In one instance, the two functions do not touch; in another instance, they touch at the end point a, and in the last instance they touch at both end points.\" width=\"963\" height=\"274\" \/> Figure 2.\u00a0A Type I region lies between two vertical lines and the graphs of two functions of\u00a0[latex]x[\/latex].[\/caption]&nbsp;\r\n\r\n[caption id=\"attachment_1321\" align=\"aligncenter\" width=\"665\"]<img class=\"size-full wp-image-1321\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/23155104\/5-2-3.jpeg\" alt=\"The graphs show a region marked D. In all instances, between c and d, there is a shape that is defined by two vertically oriented functions x = h1(y) and x = h2(y). In one instance, the two functions do not touch; in the other instance, they touch at the end point c.\" width=\"665\" height=\"310\" \/> Figure 3.\u00a0A Type II region lies between two horizontal lines and the graphs of two functions of [latex]y[\/latex].[\/caption]<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"whitespace-normal break-words\"><strong>Identifying Region Types<\/strong><\/p>\r\n<p class=\"whitespace-normal break-words\">For a <strong>Type I<\/strong> region, think vertically: identify the functions that form the top and bottom boundaries, then determine the [latex]x[\/latex]-interval.<\/p>\r\n<p class=\"whitespace-normal break-words\">For a <strong>Type II<\/strong> region, think horizontally: identify the functions that form the left and right boundaries, then determine the [latex]y[\/latex]-interval.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Consider the region in the first quadrant between the functions [latex]{y} = {\\sqrt{x}}[\/latex] and [latex]y=x^{3}[\/latex] (Figure 4). Describe the region first as Type I and then as Type II.[caption id=\"attachment_1323\" align=\"aligncenter\" width=\"503\"]<img class=\"size-full wp-image-1323\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/23155305\/5-2-4.jpeg\" alt=\"The region D is drawn between two functions, namely, y = the square root of x and y = x3.\" width=\"503\" height=\"277\" \/> Figure 4. Region [latex]D[\/latex]\u00a0can be described as Type I or as Type II.[\/caption][reveal-answer q=\"375418035\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"375418035\"]\r\n<p id=\"fs-id1167793848514\">When describing a region as Type I, we need to identify the function that lies above the region and the function that lies below the region. Here, region [latex]D[\/latex] is bounded above by [latex]{y} = {\\sqrt{x}}[\/latex] and below by [latex]y=x^{3}[\/latex] in the interval for [latex]x[\/latex] in [latex][0,1][\/latex]. Hence, as Type I, [latex]D[\/latex] is described as the set [latex]{\\left \\{(x,y) \\mid 0 \\leq x \\leq 1,x^3 \\leq y \\leq \\sqrt{x} \\right \\}}[\/latex].<\/p>\r\n<p id=\"fs-id1167793524718\">However, when describing a region as Type II, we need to identify the function that lies on the left of the region and the function that lies on the right of the region. Here, the region [latex]D[\/latex] is bounded on the left by [latex]x=y^{2}[\/latex] and on the right by [latex]x = \\sqrt[3]{y}[\/latex] in the interval for [latex]y[\/latex] in [latex][0,1][\/latex]. Hence, as Type II, [latex]D[\/latex] is described as the set [latex]{\\left \\{(x,y) \\mid 0 \\leq y \\leq 1,y^2 \\leq x \\leq \\sqrt[3]{y} \\right \\}}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<h2 data-type=\"title\">General Regions of Integration<\/h2>\n<p class=\"whitespace-normal break-words\">So far we have computed double integrals over rectangular regions. However, most regions we encounter in applications are not rectangular.<\/p>\n<p>Consider a general bounded region [latex]D[\/latex] on a plane as shown in Figure 1. Since [latex]D[\/latex] is bounded, there must exist a rectangular region [latex]R[\/latex] that encloses [latex]D[\/latex]. In other words, we can find a rectangle [latex]R[\/latex] such that [latex]D[\/latex] is a subset of [latex]R[\/latex], written as [latex]D \\subseteq R[\/latex].<\/p>\n<figure id=\"attachment_1317\" aria-describedby=\"caption-attachment-1317\" style=\"width: 417px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1317\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/23152039\/5-2-1.jpeg\" alt=\"A rectangle R with a shape D inside of it. Inside D, there is a point labeled g(x, y) = f(x, y). Outside D but still inside R, there is a point labeled g(x, y) = 0.\" width=\"417\" height=\"310\" \/><figcaption id=\"caption-attachment-1317\" class=\"wp-caption-text\">Figure 1. For a region [latex]\\small{D}[\/latex] that is a subset of\u00a0[latex]\\small{R}[\/latex], we can define a function\u00a0[latex]\\small{g(x,y)}[\/latex] to equal\u00a0[latex]\\small{f(x,y)}[\/latex] at every point in\u00a0[latex]\\small{D}[\/latex] and\u00a0[latex]\\small{0}[\/latex] at every point of\u00a0[latex]\\small{R}[\/latex] not in\u00a0[latex]\\small{D}[\/latex].<\/figcaption><\/figure>\n<p class=\"whitespace-normal break-words\">Now suppose [latex]z=f(x, y)[\/latex] is defined on this general bounded region [latex]D[\/latex]. To develop double integrals of [latex]f[\/latex] over [latex]D[\/latex], we extend the definition of the function to include all points on a rectangular region [latex]R[\/latex] that contains [latex]D[\/latex]. We define a new function [latex]g(x, y)[\/latex] on [latex]R[\/latex] as follows:<\/p>\n<p style=\"text-align: center;\">[latex]g(x,y)= \\begin{cases} f(x,y) & \\text{if } (x,y) \\text{ is in } D \\\\ 0 & \\text{if } (x,y) \\text{ is in } R \\text{ but not in } D \\end{cases}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">This extension allows us to use the integration techniques from rectangular regions. The function [latex]g(x,y)[\/latex] equals [latex]f(x,y)[\/latex] inside [latex]D[\/latex] and equals zero outside [latex]D[\/latex] but within [latex]R[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">We assume the boundary of [latex]D[\/latex] is a piecewise smooth and continuous simple closed curve to avoid technical difficulties. Under this assumption, [latex]g(x, y)[\/latex] is integrable over the rectangular region [latex]R[\/latex] as long as [latex]f(x,y)[\/latex] is integrable over [latex]D[\/latex]. For now we focus on describing different types of regions rather than the technical details of the function.<\/p>\n<p class=\"whitespace-normal break-words\">We consider two types of planar bounded regions.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>type I and type II regions<\/h3>\n<p id=\"fs-id1167793884044\">A region [latex]D[\/latex] in the [latex](x, y)[\/latex]-plane is of\u00a0<strong><span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term209\" data-type=\"term\">Type I<\/span>\u00a0<\/strong>if it lies between two vertical lines and the graphs of two continuous functions [latex]g_1(x)[\/latex] and [latex]g_2(x)[\/latex]. That is,,<\/p>\n<p style=\"text-align: center;\">[latex]{D} = \\left \\{(x,y) \\mid a \\leq x \\leq \\ b,g_1 \\leq y \\leq g_2 \\ (x) \\right \\}[\/latex].<\/p>\n<p id=\"fs-id1167794337439\">A region [latex]D[\/latex] in the [latex]xy[\/latex] plane is of\u00a0<strong><span id=\"f783ca08-ca2f-4ebb-a559-d1b30c2f9b8c_term210\" data-type=\"term\">Type II<\/span><\/strong> if it lies between two horizontal lines and the graphs of two continuous functions [latex]h_1(y)[\/latex] and [latex]h_2(y)[\/latex]. That is,<\/p>\n<p style=\"text-align: center;\">[latex]{D} = \\left \\{(x,y) \\mid c \\leq y \\leq \\ d,h_1 \\leq x \\leq h_2 \\ (y) \\right \\}[\/latex].<\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">Figures 2 and 3 illustrate the geometric interpretation of Type I and Type II regions.<\/p>\n<figure id=\"attachment_1320\" aria-describedby=\"caption-attachment-1320\" style=\"width: 963px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1320\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/23155027\/5-2-2.jpeg\" alt=\"The graphs showing a region marked D. In all instances, between a and b, there is a shape that is defined by two functions g1(x) and g2(x). In one instance, the two functions do not touch; in another instance, they touch at the end point a, and in the last instance they touch at both end points.\" width=\"963\" height=\"274\" \/><figcaption id=\"caption-attachment-1320\" class=\"wp-caption-text\">Figure 2.\u00a0A Type I region lies between two vertical lines and the graphs of two functions of\u00a0[latex]x[\/latex].<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_1321\" aria-describedby=\"caption-attachment-1321\" style=\"width: 665px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1321\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/23155104\/5-2-3.jpeg\" alt=\"The graphs show a region marked D. In all instances, between c and d, there is a shape that is defined by two vertically oriented functions x = h1(y) and x = h2(y). In one instance, the two functions do not touch; in the other instance, they touch at the end point c.\" width=\"665\" height=\"310\" \/><figcaption id=\"caption-attachment-1321\" class=\"wp-caption-text\">Figure 3.\u00a0A Type II region lies between two horizontal lines and the graphs of two functions of [latex]y[\/latex].<\/figcaption><\/figure>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"whitespace-normal break-words\"><strong>Identifying Region Types<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">For a <strong>Type I<\/strong> region, think vertically: identify the functions that form the top and bottom boundaries, then determine the [latex]x[\/latex]-interval.<\/p>\n<p class=\"whitespace-normal break-words\">For a <strong>Type II<\/strong> region, think horizontally: identify the functions that form the left and right boundaries, then determine the [latex]y[\/latex]-interval.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Consider the region in the first quadrant between the functions [latex]{y} = {\\sqrt{x}}[\/latex] and [latex]y=x^{3}[\/latex] (Figure 4). Describe the region first as Type I and then as Type II.<\/p>\n<figure id=\"attachment_1323\" aria-describedby=\"caption-attachment-1323\" style=\"width: 503px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1323\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/23155305\/5-2-4.jpeg\" alt=\"The region D is drawn between two functions, namely, y = the square root of x and y = x3.\" width=\"503\" height=\"277\" \/><figcaption id=\"caption-attachment-1323\" class=\"wp-caption-text\">Figure 4. Region [latex]D[\/latex]\u00a0can be described as Type I or as Type II.<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q375418035\">Show Solution<\/button><\/p>\n<div id=\"q375418035\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793848514\">When describing a region as Type I, we need to identify the function that lies above the region and the function that lies below the region. Here, region [latex]D[\/latex] is bounded above by [latex]{y} = {\\sqrt{x}}[\/latex] and below by [latex]y=x^{3}[\/latex] in the interval for [latex]x[\/latex] in [latex][0,1][\/latex]. Hence, as Type I, [latex]D[\/latex] is described as the set [latex]{\\left \\{(x,y) \\mid 0 \\leq x \\leq 1,x^3 \\leq y \\leq \\sqrt{x} \\right \\}}[\/latex].<\/p>\n<p id=\"fs-id1167793524718\">However, when describing a region as Type II, we need to identify the function that lies on the left of the region and the function that lies on the right of the region. Here, the region [latex]D[\/latex] is bounded on the left by [latex]x=y^{2}[\/latex] and on the right by [latex]x = \\sqrt[3]{y}[\/latex] in the interval for [latex]y[\/latex] in [latex][0,1][\/latex]. Hence, as Type II, [latex]D[\/latex] is described as the set [latex]{\\left \\{(x,y) \\mid 0 \\leq y \\leq 1,y^2 \\leq x \\leq \\sqrt[3]{y} \\right \\}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1191","chapter","type-chapter","status-publish","hentry"],"part":432,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1191","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1191\/revisions"}],"predecessor-version":[{"id":1852,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1191\/revisions\/1852"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/432"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1191\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=1191"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=1191"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=1191"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=1191"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}