{"id":672,"date":"2025-10-09T15:29:50","date_gmt":"2025-10-09T15:29:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=672"},"modified":"2025-10-15T15:48:01","modified_gmt":"2025-10-15T15:48:01","slug":"double-integration-and-applications-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/double-integration-and-applications-cheat-sheet\/","title":{"raw":"Double Integration and Applications: Cheat Sheet","rendered":"Double Integration and Applications: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Double Integrals over Basic Regions<\/strong>\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>A general bounded region [latex]D[\/latex]\u00a0on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.<\/li>\r\n \t<li>To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.<\/li>\r\n<\/ul>\r\n<strong>Applications of Double Integrals<\/strong>\r\n<ul>\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 \t<li>We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.<\/li>\r\n \t<li>We can use Fubini\u2019s theorem for improper integrals to evaluate some types of improper integrals.<\/li>\r\n<\/ul>\r\n<strong>Double Integrals in Polar Coordinates<\/strong>\r\n<ul>\r\n \t<li>To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.<\/li>\r\n \t<li>The area [latex]dA[\/latex]\u00a0in polar coordinates becomes [latex]rdrd\\theta[\/latex].<\/li>\r\n \t<li>Use [latex]x=r\\cos\\theta[\/latex], [latex]y=r\\sin\\theta[\/latex], and [latex]dA=rdrd\\theta[\/latex]\u00a0to convert an integral in rectangular coordinates to an integral in polar coordinates.<\/li>\r\n \t<li>Use [latex]r^{2}=x^{2}+y^{2}[\/latex] and [latex]\\theta=\\tan^{-1}\\left(\\frac{y}{x}\\right)[\/latex] to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.<\/li>\r\n \t<li>To find the volume in polar coordinates bounded above by a surface [latex]z=f(r,\\theta)[\/latex] over a region on the [latex]xy[\/latex]-plane, use a double integral in polar coordinates.<\/li>\r\n<\/ul>\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 \t<li><strong>Iterated integral over a Type I region\r\n<\/strong>[latex]\\displaystyle\\iint_{D} f(x,y)dA=\\displaystyle\\iint_{D} f(x,y)dydx=\\displaystyle\\int_{a}^{b}\\left[\\displaystyle\\int_{g_{1}(x)}^{g_{2}(x)}f(x,y)dy\\right] dx[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Iterated integral over a Type II region<\/strong>\r\n[latex]\\displaystyle\\iint_{D} f(x,y)dA=\\displaystyle\\iint_{D} f(x,y)dydx=\\displaystyle\\int_{c}^{d}\\left[\\displaystyle\\int_{h_{1}(y)}^{h_{2}(y)}f(x,y)dx\\right] dy[\/latex]<\/li>\r\n \t<li><strong>Double integral over a polar rectangular region\r\n<\/strong>[latex]\\underset{R}{\\displaystyle\\iint} f(r,\\theta)dA = \\underset{m,n\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f({r_{ij}}^{\\ast},{\\theta_{ij}}^{\\ast})\\Delta{A} =\\underset{m,n\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f({r_{ij}}^{\\ast},{\\theta_{ij}}^{\\ast}){r_{ij}}^{\\ast}\\Delta{r}\\Delta{\\theta}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Double integral over a general polar region<\/strong><strong>\r\n<\/strong>[latex]\\underset{D}{\\displaystyle\\iint} f(r,\\theta)r dr d\\theta=\\displaystyle\\int_{\\theta=\\alpha}^{\\theta=\\beta} \\displaystyle\\int_{r=h_{1}(\\theta)}^{r=h_{2}(\\theta)}f(r,\\theta)r dr d\\theta[\/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]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>improper double integral<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a double integral over an unbounded region or of an unbounded function<\/span><\/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>\r\n<dl class=\"definition\">\r\n \t<dt>polar rectangle<\/dt>\r\n \t<dd>the region enclosed between the circles [latex]r=a[\/latex] and [latex]r=b[\/latex]\u00a0and the angles [latex]\\theta = \\alpha[\/latex]\u00a0and [latex]\\theta = \\beta[\/latex]; it is described as\u00a0[latex]{\\bf{R}}=\\{(r,{\\theta}) | a{\\leq}r{\\leq}b, {\\alpha}{\\leq}{\\theta}{\\leq}{\\beta}\\}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>Type I<\/dt>\r\n \t<dd>a region\u00a0[latex]\\bf{D}[\/latex] in the [latex]xy[\/latex]-plane is Type I\u00a0if it lies between two vertical lines and the graphs of two continuous functions [latex]g_{1}(x)[\/latex] and\u00a0[latex]g_{2}(x)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>Type II<\/dt>\r\n \t<dd>a region [latex]\\bf{D}[\/latex]<strong>\u00a0<\/strong>in the [latex]xy[\/latex]-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions [latex]h_{1}(y)[\/latex]\u00a0and\u00a0[latex]h_{2}(y)[\/latex]<\/dd>\r\n<\/dl>","rendered":"<h2>Essential Concepts<\/h2>\n<p><strong>Double Integrals over Basic Regions<\/strong><\/p>\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>A general bounded region [latex]D[\/latex]\u00a0on the plane is a region that can be enclosed inside a rectangular region. We can use this idea to define a double integral over a general bounded region.<\/li>\n<li>To evaluate an iterated integral of a function over a general nonrectangular region, we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.<\/li>\n<\/ul>\n<p><strong>Applications of Double Integrals<\/strong><\/p>\n<ul>\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<li>We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.<\/li>\n<li>We can use Fubini\u2019s theorem for improper integrals to evaluate some types of improper integrals.<\/li>\n<\/ul>\n<p><strong>Double Integrals in Polar Coordinates<\/strong><\/p>\n<ul>\n<li>To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.<\/li>\n<li>The area [latex]dA[\/latex]\u00a0in polar coordinates becomes [latex]rdrd\\theta[\/latex].<\/li>\n<li>Use [latex]x=r\\cos\\theta[\/latex], [latex]y=r\\sin\\theta[\/latex], and [latex]dA=rdrd\\theta[\/latex]\u00a0to convert an integral in rectangular coordinates to an integral in polar coordinates.<\/li>\n<li>Use [latex]r^{2}=x^{2}+y^{2}[\/latex] and [latex]\\theta=\\tan^{-1}\\left(\\frac{y}{x}\\right)[\/latex] to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.<\/li>\n<li>To find the volume in polar coordinates bounded above by a surface [latex]z=f(r,\\theta)[\/latex] over a region on the [latex]xy[\/latex]-plane, use a double integral in polar coordinates.<\/li>\n<\/ul>\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<li><strong>Iterated integral over a Type I region<br \/>\n<\/strong>[latex]\\displaystyle\\iint_{D} f(x,y)dA=\\displaystyle\\iint_{D} f(x,y)dydx=\\displaystyle\\int_{a}^{b}\\left[\\displaystyle\\int_{g_{1}(x)}^{g_{2}(x)}f(x,y)dy\\right] dx[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Iterated integral over a Type II region<\/strong><br \/>\n[latex]\\displaystyle\\iint_{D} f(x,y)dA=\\displaystyle\\iint_{D} f(x,y)dydx=\\displaystyle\\int_{c}^{d}\\left[\\displaystyle\\int_{h_{1}(y)}^{h_{2}(y)}f(x,y)dx\\right] dy[\/latex]<\/li>\n<li><strong>Double integral over a polar rectangular region<br \/>\n<\/strong>[latex]\\underset{R}{\\displaystyle\\iint} f(r,\\theta)dA = \\underset{m,n\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f({r_{ij}}^{\\ast},{\\theta_{ij}}^{\\ast})\\Delta{A} =\\underset{m,n\\to\\infty}{\\lim}\\displaystyle\\sum_{i=1}^{m}\\displaystyle\\sum_{j=1}^{n}f({r_{ij}}^{\\ast},{\\theta_{ij}}^{\\ast}){r_{ij}}^{\\ast}\\Delta{r}\\Delta{\\theta}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Double integral over a general polar region<\/strong><strong><br \/>\n<\/strong>[latex]\\underset{D}{\\displaystyle\\iint} f(r,\\theta)r dr d\\theta=\\displaystyle\\int_{\\theta=\\alpha}^{\\theta=\\beta} \\displaystyle\\int_{r=h_{1}(\\theta)}^{r=h_{2}(\\theta)}f(r,\\theta)r dr d\\theta[\/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]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>improper double integral<\/dt>\n<dd><span style=\"font-size: 1em;\">a double integral over an unbounded region or of an unbounded function<\/span><\/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<dl class=\"definition\">\n<dt>polar rectangle<\/dt>\n<dd>the region enclosed between the circles [latex]r=a[\/latex] and [latex]r=b[\/latex]\u00a0and the angles [latex]\\theta = \\alpha[\/latex]\u00a0and [latex]\\theta = \\beta[\/latex]; it is described as\u00a0[latex]{\\bf{R}}=\\{(r,{\\theta}) | a{\\leq}r{\\leq}b, {\\alpha}{\\leq}{\\theta}{\\leq}{\\beta}\\}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Type I<\/dt>\n<dd>a region\u00a0[latex]\\bf{D}[\/latex] in the [latex]xy[\/latex]-plane is Type I\u00a0if it lies between two vertical lines and the graphs of two continuous functions [latex]g_{1}(x)[\/latex] and\u00a0[latex]g_{2}(x)[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Type II<\/dt>\n<dd>a region [latex]\\bf{D}[\/latex]<strong>\u00a0<\/strong>in the [latex]xy[\/latex]-plane is Type II if it lies between two horizontal lines and the graphs of two continuous functions [latex]h_{1}(y)[\/latex]\u00a0and\u00a0[latex]h_{2}(y)[\/latex]<\/dd>\n<\/dl>\n","protected":false},"author":15,"menu_order":1,"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-672","chapter","type-chapter","status-publish","hentry"],"part":432,"module-header":"cheat_sheet","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\/672","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":4,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/672\/revisions"}],"predecessor-version":[{"id":1246,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/672\/revisions\/1246"}],"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\/672\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=672"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=672"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=672"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=672"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}