{"id":132,"date":"2021-07-30T17:19:04","date_gmt":"2021-07-30T17:19:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=132"},"modified":"2022-11-01T04:16:06","modified_gmt":"2022-11-01T04:16:06","slug":"summary-of-double-integrals-over-rectangular-regions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-double-integrals-over-rectangular-regions\/","title":{"raw":"Summary of Double Integrals over Rectangular Regions","rendered":"Summary of Double Integrals over Rectangular Regions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>We can use a double Riemann sum to approximate the volume of a solid bounded above by a function of two variables over a rectangular region. By taking the limit, this becomes a double integral representing the volume of the solid.<\/li>\r\n \t<li>Properties of double integral are useful to simplify computation and find bounds on their values.<\/li>\r\n \t<li>We can use Fubini\u2019s theorem to write and evaluate a double integral as an iterated integral.<\/li>\r\n \t<li>Double integrals are used to calculate the area of a region, the volume under a surface, and the average value of a function of two variables over a rectangular region.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Double integral\r\n<\/strong>[latex]\\underset{R}{\\displaystyle\\iint} f(x,y)dA=\\underset{m,n\\to{\\infty}}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}[\/latex]<\/li>\r\n \t<li><strong>Iterated integral<\/strong>\r\n[latex]\\displaystyle\\int_{a}^{b}\\displaystyle\\int_{c}^{d} f(x,y)dxdy=\\displaystyle\\int_{a}^{b}\\left[\\displaystyle\\int_{c}^{d}f(x,y)dy\\right]dx[\/latex]\u00a0 OR\r\n[latex]\\displaystyle\\int_{c}^{d}\\displaystyle\\int_{b}^{a} f(x,y)dxdy=\\displaystyle\\int_{c}^{d}\\left[\\displaystyle\\int_{a}^{b}f(x,y)dx\\right]dy[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Average value of a function of two variables<\/strong>\r\n[latex]f_{\\text{ave}}=\\frac{1}{\\text{Area R}}\\underset{R}{\\displaystyle\\iint} f(x,y)dxdy[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>double Riemann Sum<\/dt>\r\n \t<dd>of the function [latex]f(x,y)[\/latex]\u00a0over\u00a0a rectangular region [latex]R[\/latex]\u00a0is [latex]\\displaystyle\\sum_{i=1}^{m} {} \\displaystyle\\sum_{j=1}^{n} {f({x^{*}}_{i,j}, {y^{*}}_{i,j})}[\/latex] where [latex]R[\/latex]\u00a0is divided into smaller sub rectangles [latex]R_{ij}[\/latex] and [latex]({x^{*}}_{i,j}, {y^{*}}_{i,j})[\/latex]\u00a0is an arbitrary point in\u00a0[latex]R_{ij}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>double Integral<\/dt>\r\n \t<dd>of the function [latex]f(x,y)[\/latex]<strong>\u00a0<\/strong>over the region [latex]R[\/latex]\u00a0in the [latex]xy[\/latex]-plane\u00a0is defined as the limit of a double Riemann sum,\u00a0[latex]\\underset{R}{\\displaystyle\\iint} f(x,y)dA=\\underset{m,n\\to{\\infty}}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>Fubini's Theorem<\/dt>\r\n \t<dd>if\u00a0[latex]f(x,y)[\/latex] is a function of two variables that is continuous over a rectangular region [latex]R = \\{(x,y)\\in{\\mathbb{R}}^{2}|a\\leq x\\leq b,c\\leq y\\leq d\\}[\/latex], then the double integral of [latex]f[\/latex] over the region equals an iterated integral,<\/dd>\r\n \t<dd>[latex]\\underset{R}{\\displaystyle\\iint} f(x,y)dxdy={\\displaystyle\\int_{a}^{b}}{\\displaystyle\\int_{c}^{d} {f(x,y){dx}{dy}}}={\\displaystyle\\int_{c}^{d}}{\\displaystyle\\int_{a}^{b} {f(x,y){dx}{dy}}}[\/latex]\r\n<dl class=\"definition\"><\/dl>\r\n<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>iterated Integral<\/dt>\r\n \t<dd>for a function [latex]f(x,y)[\/latex] over the region [latex]\\bf{R}[\/latex] is\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">[latex]{\\displaystyle\\int_{a}^{b}}{\\displaystyle\\int_{c}^{d} {f(x,y){dx}{dy}}}={\\displaystyle\\int_{a}^{b}}\\left[{\\displaystyle\\int_{c}^{d} {f(x,y){dy}}}\\right]{dx}[\/latex]<\/span>\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">[latex]{\\displaystyle\\int_{a}^{b}}{\\displaystyle\\int_{c}^{d} {f(x,y){dx}{dy}}}={\\displaystyle\\int_{c}^{d}}\\left[{\\displaystyle\\int_{a}^{b} {f(x,y){dx}}}\\right]{dy}[\/latex]<\/span><\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>We can use a double Riemann sum to approximate the volume of a solid bounded above by a function of two variables over a rectangular region. By taking the limit, this becomes a double integral representing the volume of the solid.<\/li>\n<li>Properties of double integral are useful to simplify computation and find bounds on their values.<\/li>\n<li>We can use Fubini\u2019s theorem to write and evaluate a double integral as an iterated integral.<\/li>\n<li>Double integrals are used to calculate the area of a region, the volume under a surface, and the average value of a function of two variables over a rectangular region.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Double integral<br \/>\n<\/strong>[latex]\\underset{R}{\\displaystyle\\iint} f(x,y)dA=\\underset{m,n\\to{\\infty}}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}[\/latex]<\/li>\n<li><strong>Iterated integral<\/strong><br \/>\n[latex]\\displaystyle\\int_{a}^{b}\\displaystyle\\int_{c}^{d} f(x,y)dxdy=\\displaystyle\\int_{a}^{b}\\left[\\displaystyle\\int_{c}^{d}f(x,y)dy\\right]dx[\/latex]\u00a0 OR<br \/>\n[latex]\\displaystyle\\int_{c}^{d}\\displaystyle\\int_{b}^{a} f(x,y)dxdy=\\displaystyle\\int_{c}^{d}\\left[\\displaystyle\\int_{a}^{b}f(x,y)dx\\right]dy[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Average value of a function of two variables<\/strong><br \/>\n[latex]f_{\\text{ave}}=\\frac{1}{\\text{Area R}}\\underset{R}{\\displaystyle\\iint} f(x,y)dxdy[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>double Riemann Sum<\/dt>\n<dd>of the function [latex]f(x,y)[\/latex]\u00a0over\u00a0a rectangular region [latex]R[\/latex]\u00a0is [latex]\\displaystyle\\sum_{i=1}^{m} {} \\displaystyle\\sum_{j=1}^{n} {f({x^{*}}_{i,j}, {y^{*}}_{i,j})}[\/latex] where [latex]R[\/latex]\u00a0is divided into smaller sub rectangles [latex]R_{ij}[\/latex] and [latex]({x^{*}}_{i,j}, {y^{*}}_{i,j})[\/latex]\u00a0is an arbitrary point in\u00a0[latex]R_{ij}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>double Integral<\/dt>\n<dd>of the function [latex]f(x,y)[\/latex]<strong>\u00a0<\/strong>over the region [latex]R[\/latex]\u00a0in the [latex]xy[\/latex]-plane\u00a0is defined as the limit of a double Riemann sum,\u00a0[latex]\\underset{R}{\\displaystyle\\iint} f(x,y)dA=\\underset{m,n\\to{\\infty}}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f(x_{ij}^{\\ast},y_{ij}^{\\ast})\\Delta{A}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Fubini&#8217;s Theorem<\/dt>\n<dd>if\u00a0[latex]f(x,y)[\/latex] is a function of two variables that is continuous over a rectangular region [latex]R = \\{(x,y)\\in{\\mathbb{R}}^{2}|a\\leq x\\leq b,c\\leq y\\leq d\\}[\/latex], then the double integral of [latex]f[\/latex] over the region equals an iterated integral,<\/dd>\n<dd>[latex]\\underset{R}{\\displaystyle\\iint} f(x,y)dxdy={\\displaystyle\\int_{a}^{b}}{\\displaystyle\\int_{c}^{d} {f(x,y){dx}{dy}}}={\\displaystyle\\int_{c}^{d}}{\\displaystyle\\int_{a}^{b} {f(x,y){dx}{dy}}}[\/latex]<\/p>\n<dl class=\"definition\"><\/dl>\n<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>iterated Integral<\/dt>\n<dd>for a function [latex]f(x,y)[\/latex] over the region [latex]\\bf{R}[\/latex] is<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">[latex]{\\displaystyle\\int_{a}^{b}}{\\displaystyle\\int_{c}^{d} {f(x,y){dx}{dy}}}={\\displaystyle\\int_{a}^{b}}\\left[{\\displaystyle\\int_{c}^{d} {f(x,y){dy}}}\\right]{dx}[\/latex]<\/span><\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">[latex]{\\displaystyle\\int_{a}^{b}}{\\displaystyle\\int_{c}^{d} {f(x,y){dx}{dy}}}={\\displaystyle\\int_{c}^{d}}\\left[{\\displaystyle\\int_{a}^{b} {f(x,y){dx}}}\\right]{dy}[\/latex]<\/span><\/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-132\">\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":6,"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-132","chapter","type-chapter","status-publish","hentry"],"part":23,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/132","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":21,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/132\/revisions"}],"predecessor-version":[{"id":6451,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/132\/revisions\/6451"}],"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\/132\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=132"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=132"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=132"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}