{"id":1189,"date":"2025-10-15T15:16:38","date_gmt":"2025-10-15T15:16:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=1189"},"modified":"2025-10-29T15:18:24","modified_gmt":"2025-10-29T15:18:24","slug":"double-integrals-over-basic-regions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/double-integrals-over-basic-regions-learn-it-2\/","title":{"raw":"Double Integrals over Basic Regions: Learn It 2","rendered":"Double Integrals over Basic Regions: Learn It 2"},"content":{"raw":"<h2 data-type=\"title\">Properties of Double Integrals<\/h2>\r\n<p class=\"whitespace-normal break-words\">The properties of double integrals are very helpful when computing them or otherwise working with them. We list here six properties of double integrals. Properties 1 and 2 are referred to as the <strong>linearity<\/strong> of the integral, property 3 is the <strong>additivity<\/strong> of the integral, property 4 is the <strong>monotonicity<\/strong> of the integral, and property 5 is used to find the bounds of the integral. Property 6 is used if [latex]f(x,y)[\/latex] is a product of two functions [latex]g(x)[\/latex] and [latex]h(y)[\/latex].<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>theorem: properties of double integrals<\/h3>\r\n<p class=\"whitespace-normal break-words\">Assume that the functions [latex]f(x,y)[\/latex] and [latex]g(x,y)[\/latex] are integrable over the rectangular region [latex]R[\/latex]; [latex]S[\/latex] and [latex]T[\/latex] are subregions of [latex]R[\/latex]; and assume that [latex]m[\/latex] and [latex]M[\/latex] are real numbers.<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">The sum [latex]f(x,y) + g(x,y)[\/latex] is integrable and:\r\n<center>[latex]\\underset{R}{\\displaystyle\\iint}[f(x,y)+g(x,y)]dA=\\underset{R}{\\displaystyle\\iint}f(x,y)dA+\\underset{R}{\\displaystyle\\iint}g(x,y)dA[\/latex]<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]c[\/latex] is a constant, then [latex]cf(x,y)[\/latex] is integrable and:\r\n<center>[latex]\\underset{R}{\\displaystyle\\iint}cf(x,y)dA = c\\underset{R}{\\displaystyle\\iint}f(x,y)dA[\/latex]<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]R = S \\cup T[\/latex] and [latex]S \\cap T = \\emptyset[\/latex] except an overlap on the boundaries, then:\r\n<center>[latex]\\underset{R}{\\displaystyle\\iint}f(x,y)dA = \\underset{S}{\\displaystyle\\iint}f(x,y)dA + \\underset{T}{\\displaystyle\\iint}f(x,y)dA[\/latex]<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]f(x,y)\\geq g(x,y)[\/latex] for [latex](x,y)[\/latex] in [latex]R[\/latex], then:\r\n<center>[latex]\\underset{R}{\\displaystyle\\iint}f(x,y)dA\\geq \\underset{R}{\\displaystyle\\iint}g(x,y)dA[\/latex]<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">If [latex]m \\leq f(x,y)\\leq M[\/latex], then:\r\n<center>[latex]m \\times A(R) \\leq \\underset{R}{\\displaystyle\\iint}f(x,y)dA \\leq M \\times A(R)[\/latex]<\/center><\/li>\r\n \t<li class=\"whitespace-normal break-words\">In the case where [latex]f(x,y)[\/latex] can be factored as a product of a function [latex]g(x)[\/latex] of [latex]x[\/latex] only and a function [latex]h(y)[\/latex] of [latex]y[\/latex] only, then over the region [latex]R = { (x,y)\\mid a \\leq x \\leq b,c \\leq y \\leq d }[\/latex], the double integral can be written as:\r\n<center>[latex]\\underset{R}{\\displaystyle\\iint}f(x,y)dA = \\left(\\displaystyle\\int_a^bg(x)dx\\right)\\left(\\displaystyle\\int_c^dh(y)dy\\right)[\/latex]<\/center><\/li>\r\n<\/ol>\r\n<\/section>These properties are used in the evaluation of double integrals, as we will see later.","rendered":"<h2 data-type=\"title\">Properties of Double Integrals<\/h2>\n<p class=\"whitespace-normal break-words\">The properties of double integrals are very helpful when computing them or otherwise working with them. We list here six properties of double integrals. Properties 1 and 2 are referred to as the <strong>linearity<\/strong> of the integral, property 3 is the <strong>additivity<\/strong> of the integral, property 4 is the <strong>monotonicity<\/strong> of the integral, and property 5 is used to find the bounds of the integral. Property 6 is used if [latex]f(x,y)[\/latex] is a product of two functions [latex]g(x)[\/latex] and [latex]h(y)[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>theorem: properties of double integrals<\/h3>\n<p class=\"whitespace-normal break-words\">Assume that the functions [latex]f(x,y)[\/latex] and [latex]g(x,y)[\/latex] are integrable over the rectangular region [latex]R[\/latex]; [latex]S[\/latex] and [latex]T[\/latex] are subregions of [latex]R[\/latex]; and assume that [latex]m[\/latex] and [latex]M[\/latex] are real numbers.<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-2.5 pl-7\">\n<li class=\"whitespace-normal break-words\">The sum [latex]f(x,y) + g(x,y)[\/latex] is integrable and:\n<div style=\"text-align: center;\">[latex]\\underset{R}{\\displaystyle\\iint}[f(x,y)+g(x,y)]dA=\\underset{R}{\\displaystyle\\iint}f(x,y)dA+\\underset{R}{\\displaystyle\\iint}g(x,y)dA[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]c[\/latex] is a constant, then [latex]cf(x,y)[\/latex] is integrable and:\n<div style=\"text-align: center;\">[latex]\\underset{R}{\\displaystyle\\iint}cf(x,y)dA = c\\underset{R}{\\displaystyle\\iint}f(x,y)dA[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]R = S \\cup T[\/latex] and [latex]S \\cap T = \\emptyset[\/latex] except an overlap on the boundaries, then:\n<div style=\"text-align: center;\">[latex]\\underset{R}{\\displaystyle\\iint}f(x,y)dA = \\underset{S}{\\displaystyle\\iint}f(x,y)dA + \\underset{T}{\\displaystyle\\iint}f(x,y)dA[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]f(x,y)\\geq g(x,y)[\/latex] for [latex](x,y)[\/latex] in [latex]R[\/latex], then:\n<div style=\"text-align: center;\">[latex]\\underset{R}{\\displaystyle\\iint}f(x,y)dA\\geq \\underset{R}{\\displaystyle\\iint}g(x,y)dA[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">If [latex]m \\leq f(x,y)\\leq M[\/latex], then:\n<div style=\"text-align: center;\">[latex]m \\times A(R) \\leq \\underset{R}{\\displaystyle\\iint}f(x,y)dA \\leq M \\times A(R)[\/latex]<\/div>\n<\/li>\n<li class=\"whitespace-normal break-words\">In the case where [latex]f(x,y)[\/latex] can be factored as a product of a function [latex]g(x)[\/latex] of [latex]x[\/latex] only and a function [latex]h(y)[\/latex] of [latex]y[\/latex] only, then over the region [latex]R = { (x,y)\\mid a \\leq x \\leq b,c \\leq y \\leq d }[\/latex], the double integral can be written as:\n<div style=\"text-align: center;\">[latex]\\underset{R}{\\displaystyle\\iint}f(x,y)dA = \\left(\\displaystyle\\int_a^bg(x)dx\\right)\\left(\\displaystyle\\int_c^dh(y)dy\\right)[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/section>\n<p>These properties are used in the evaluation of double integrals, as we will see later.<\/p>\n","protected":false},"author":15,"menu_order":7,"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-1189","chapter","type-chapter","status-publish","hentry"],"part":432,"module-header":"- Select Header 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