{"id":677,"date":"2025-10-09T15:29:37","date_gmt":"2025-10-09T15:29:37","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=677"},"modified":"2025-12-12T13:46:24","modified_gmt":"2025-12-12T13:46:24","slug":"double-integrals-over-basic-regions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/double-integrals-over-basic-regions-apply-it\/","title":{"raw":"Double Integrals over Basic Regions: Apply It","rendered":"Double Integrals over Basic Regions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Calculate double integrals when your region is a rectangle<\/li>\r\n \t<li>Find double integrals over more complex regions<\/li>\r\n \t<li>Switch the order of integration to make calculations easier<\/li>\r\n<\/ul>\r\n<\/section>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid gap-4 [&amp;_&gt;_*]:min-w-0\">\r\n<h2 class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Calculating Material Needs for a Shelter Floor<\/h2>\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"standard-markdown grid-cols-1 grid gap-4 [&amp;_&gt;_*]:min-w-0\">You are volunteering with a disaster relief organization that builds temporary shelters. Your task is to calculate the amount of insulating foam needed for the floor of a new shelter. The shelter floor is rectangular, measuring [latex]5[\/latex] meters by [latex]4[\/latex] meters. Due to the uneven ground beneath, the foam thickness varies across the floor according to the function [latex]f(x,y) = 6 - x - y[\/latex] centimeters, where [latex]x[\/latex] is the distance in meters from the west edge and [latex]y[\/latex] is the distance in meters from the south edge. To determine the total volume of foam required, you need to evaluate a double integral over this region. Later, you'll also help calculate materials for a triangular storage area where the integration region is not rectangular.<\/div>\r\n<div><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]316949[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]316950[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]316951[\/ohm_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1] 316952[\/ohm_question]<\/section><\/div>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Calculate double integrals when your region is a rectangle<\/li>\n<li>Find double integrals over more complex regions<\/li>\n<li>Switch the order of integration to make calculations easier<\/li>\n<\/ul>\n<\/section>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid gap-4 [&amp;_&gt;_*]:min-w-0\">\n<h2 class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Calculating Material Needs for a Shelter Floor<\/h2>\n<\/div>\n<\/div>\n<div>\n<div class=\"standard-markdown grid-cols-1 grid gap-4 [&amp;_&gt;_*]:min-w-0\">You are volunteering with a disaster relief organization that builds temporary shelters. Your task is to calculate the amount of insulating foam needed for the floor of a new shelter. The shelter floor is rectangular, measuring [latex]5[\/latex] meters by [latex]4[\/latex] meters. Due to the uneven ground beneath, the foam thickness varies across the floor according to the function [latex]f(x,y) = 6 - x - y[\/latex] centimeters, where [latex]x[\/latex] is the distance in meters from the west edge and [latex]y[\/latex] is the distance in meters from the south edge. To determine the total volume of foam required, you need to evaluate a double integral over this region. Later, you&#8217;ll also help calculate materials for a triangular storage area where the integration region is not rectangular.<\/div>\n<div>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm316949\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316949&theme=lumen&iframe_resize_id=ohm316949&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm316950\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316950&theme=lumen&iframe_resize_id=ohm316950&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm316951\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316951&theme=lumen&iframe_resize_id=ohm316951&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm316952\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=316952&theme=lumen&iframe_resize_id=ohm316952&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n<\/div>\n","protected":false},"author":15,"menu_order":12,"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-677","chapter","type-chapter","status-publish","hentry"],"part":432,"module-header":"apply_it","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\/677","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":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/677\/revisions"}],"predecessor-version":[{"id":2773,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/677\/revisions\/2773"}],"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\/677\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=677"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=677"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=677"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=677"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}