{"id":441,"date":"2025-02-13T19:44:56","date_gmt":"2025-02-13T19:44:56","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/volumes-of-revolution-cylindrical-shells-fresh-take\/"},"modified":"2025-02-13T19:44:56","modified_gmt":"2025-02-13T19:44:56","slug":"volumes-of-revolution-cylindrical-shells-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/volumes-of-revolution-cylindrical-shells-fresh-take\/","title":{"raw":"Volumes of Revolution: Cylindrical Shells: Fresh Take","rendered":"Volumes of Revolution: Cylindrical Shells: Fresh Take"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Determine the volume of a solid formed by rotating a region around an axis using cylindrical shells<\/li>\n\t<li>Evaluate the benefits and limitations of different methods (disk, washer, cylindrical shells) for calculating volumes<\/li>\n<\/ul>\n<\/section>\n<h2>Cylindrical Shells Method<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Cylindrical Shells Method:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Used for solids of revolution, especially when integrating parallel to the axis of rotation<\/li>\n\t<li class=\"whitespace-normal break-words\">Key formula for rotation around [latex]y[\/latex]-axis:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]V = \\int_a^b 2\\pi x f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Key formula for rotation around [latex]x[\/latex]-axis:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">[latex]V = \\int_c^d 2\\pi y g(y) dy[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Adaptability:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Can be modified for rotation around lines other than coordinate axes<\/li>\n\t<li class=\"whitespace-normal break-words\">For rotation around [latex]x = -k[\/latex]: [latex]V = \\int_a^b 2\\pi (x+k) f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Comparison with Other Methods:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Disk Method: Used when rotating around an axis perpendicular to the rectangles<\/li>\n\t<li class=\"whitespace-normal break-words\">Washer Method: Similar to disk method, but for solids with cavities<\/li>\n\t<li class=\"whitespace-normal break-words\">Choice depends on ease of integration and problem setup<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Advantages:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Often simplifies calculations for [latex]y[\/latex]-axis rotations of [latex]x[\/latex]-defined functions<\/li>\n\t<li class=\"whitespace-normal break-words\">Useful when other methods require multiple integrals<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Key Concepts<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Shell Volume:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Approximated by [latex]V_{shell} \\approx 2\\pi x^* f(x^*) \\Delta x[\/latex] for y-axis rotation<\/li>\n\t<li class=\"whitespace-normal break-words\">Height of shell is the function value, radius is the [latex]x[\/latex]-coordinate<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Axis of Rotation:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Formula adjusts based on the axis of rotation<\/li>\n\t<li class=\"whitespace-normal break-words\">Radius term in integral changes for different axes<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Multiple Functions:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">For regions bounded by two functions, use [latex]f(x) - g(x)[\/latex] for shell height<\/li>\n<\/ul>\n<\/li>\n\t<li class=\"whitespace-normal break-words\">Method Selection:\n\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n\t<li class=\"whitespace-normal break-words\">Consider the axis of rotation and how functions are defined<\/li>\n\t<li class=\"whitespace-normal break-words\">Choose method that results in simplest integration<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p>Define [latex]R[\/latex]&nbsp;as the region bounded above by the graph of [latex]f(x)={x}^{2}[\/latex] and below by the [latex]x[\/latex]-axis over the interval [latex]\\left[1,2\\right].[\/latex] Find the volume of the solid of revolution formed by revolving [latex]R[\/latex] around the [latex]y\\text{-axis}.[\/latex]<\/p>\n<p>[reveal-answer q=\"2007633\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"2007633\"]<\/p>\n<p id=\"fs-id1167793414077\">[latex]\\frac{15\\pi }{2}[\/latex] units<sup>3<\/sup><\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Define [latex]R[\/latex] as the region bounded above by the graph of [latex]f(x)=3x-{x}^{2}[\/latex] and below by the [latex]x\\text{-axis}[\/latex] over the interval [latex]\\left[0,2\\right].[\/latex] Find the volume of the solid of revolution formed by revolving [latex]R[\/latex] around the [latex]y\\text{-axis}.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1167793473593\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793473593\"]\n\n<p id=\"fs-id1167793473593\">[latex]8\\pi [\/latex] units<sup>3<\/sup><\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Define [latex]Q[\/latex] as the region bounded on the right by the graph of [latex]g(y)=3\\text{\/}y[\/latex] and on the left by the [latex]y\\text{-axis}[\/latex] for [latex]y\\in \\left[1,3\\right].[\/latex] Find the volume of the solid of revolution formed by revolving [latex]Q[\/latex] around the [latex]x\\text{-axis}.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1167793466686\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793466686\"]\n\n<p id=\"fs-id1167793466686\">[latex]12\\pi [\/latex] units<sup>3<\/sup><\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/3Rq70sJECwQ?controls=0&amp;start=554&amp;end=696&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.3VolumesOfRevolutionCylindricalShells554to696_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.3 Volumes of Revolution: Cylindrical Shells\" here (opens in new window).<\/a>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Define [latex]R[\/latex] as the region bounded above by the graph of [latex]f(x)={x}^{2}[\/latex] and below by the [latex]x\\text{-axis}[\/latex] over the interval [latex]\\left[0,1\\right].[\/latex] Find the volume of the solid of revolution formed by revolving [latex]R[\/latex] around the line [latex]x=-2.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1167794138246\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167794138246\"]\n\n<p id=\"fs-id1167794138246\">[latex]\\frac{11\\pi }{6}[\/latex] units<sup>3<\/sup><\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/3Rq70sJECwQ?controls=0&amp;start=1099&amp;end=1246&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.3VolumesOfRevolutionCylindricalShells1099to1246_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"2.3 Volumes of Revolution: Cylindrical Shells\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Define [latex]R[\/latex] as the region bounded above by the graph of [latex]f(x)=x[\/latex] and below by the graph of [latex]g(x)={x}^{2}[\/latex] over the interval [latex]\\left[0,1\\right].[\/latex] Find the volume of the solid of revolution formed by revolving [latex]R[\/latex] around the [latex]y\\text{-axis}.[\/latex]<\/p>\n\n[reveal-answer q=\"fs-id1167793691567\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"fs-id1167793691567\"]\n\n<p id=\"fs-id1167793691567\">[latex]\\frac{\\pi }{6}[\/latex] units<sup>3<\/sup><\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Determine the volume of a solid formed by rotating a region around an axis using cylindrical shells<\/li>\n<li>Evaluate the benefits and limitations of different methods (disk, washer, cylindrical shells) for calculating volumes<\/li>\n<\/ul>\n<\/section>\n<h2>Cylindrical Shells Method<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea&nbsp;<\/strong><\/p>\n<ul class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Cylindrical Shells Method:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Used for solids of revolution, especially when integrating parallel to the axis of rotation<\/li>\n<li class=\"whitespace-normal break-words\">Key formula for rotation around [latex]y[\/latex]-axis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]V = \\int_a^b 2\\pi x f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key formula for rotation around [latex]x[\/latex]-axis:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]V = \\int_c^d 2\\pi y g(y) dy[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Adaptability:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Can be modified for rotation around lines other than coordinate axes<\/li>\n<li class=\"whitespace-normal break-words\">For rotation around [latex]x = -k[\/latex]: [latex]V = \\int_a^b 2\\pi (x+k) f(x) dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Comparison with Other Methods:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Disk Method: Used when rotating around an axis perpendicular to the rectangles<\/li>\n<li class=\"whitespace-normal break-words\">Washer Method: Similar to disk method, but for solids with cavities<\/li>\n<li class=\"whitespace-normal break-words\">Choice depends on ease of integration and problem setup<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Advantages:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Often simplifies calculations for [latex]y[\/latex]-axis rotations of [latex]x[\/latex]-defined functions<\/li>\n<li class=\"whitespace-normal break-words\">Useful when other methods require multiple integrals<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-bold\"><strong>Key Concepts<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Shell Volume:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Approximated by [latex]V_{shell} \\approx 2\\pi x^* f(x^*) \\Delta x[\/latex] for y-axis rotation<\/li>\n<li class=\"whitespace-normal break-words\">Height of shell is the function value, radius is the [latex]x[\/latex]-coordinate<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Axis of Rotation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Formula adjusts based on the axis of rotation<\/li>\n<li class=\"whitespace-normal break-words\">Radius term in integral changes for different axes<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Multiple Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For regions bounded by two functions, use [latex]f(x) - g(x)[\/latex] for shell height<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Method Selection:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Consider the axis of rotation and how functions are defined<\/li>\n<li class=\"whitespace-normal break-words\">Choose method that results in simplest integration<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">\n<p>Define [latex]R[\/latex]&nbsp;as the region bounded above by the graph of [latex]f(x)={x}^{2}[\/latex] and below by the [latex]x[\/latex]-axis over the interval [latex]\\left[1,2\\right].[\/latex] Find the volume of the solid of revolution formed by revolving [latex]R[\/latex] around the [latex]y\\text{-axis}.[\/latex]<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q2007633\">Show Solution<\/button><\/p>\n<div id=\"q2007633\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793414077\">[latex]\\frac{15\\pi }{2}[\/latex] units<sup>3<\/sup><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Define [latex]R[\/latex] as the region bounded above by the graph of [latex]f(x)=3x-{x}^{2}[\/latex] and below by the [latex]x\\text{-axis}[\/latex] over the interval [latex]\\left[0,2\\right].[\/latex] Find the volume of the solid of revolution formed by revolving [latex]R[\/latex] around the [latex]y\\text{-axis}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793473593\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793473593\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793473593\">[latex]8\\pi[\/latex] units<sup>3<\/sup><\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Define [latex]Q[\/latex] as the region bounded on the right by the graph of [latex]g(y)=3\\text{\/}y[\/latex] and on the left by the [latex]y\\text{-axis}[\/latex] for [latex]y\\in \\left[1,3\\right].[\/latex] Find the volume of the solid of revolution formed by revolving [latex]Q[\/latex] around the [latex]x\\text{-axis}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793466686\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793466686\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793466686\">[latex]12\\pi[\/latex] units<sup>3<\/sup><\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/3Rq70sJECwQ?controls=0&amp;start=554&amp;end=696&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.3VolumesOfRevolutionCylindricalShells554to696_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.3 Volumes of Revolution: Cylindrical Shells&#8221; here (opens in new window).<\/a><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Define [latex]R[\/latex] as the region bounded above by the graph of [latex]f(x)={x}^{2}[\/latex] and below by the [latex]x\\text{-axis}[\/latex] over the interval [latex]\\left[0,1\\right].[\/latex] Find the volume of the solid of revolution formed by revolving [latex]R[\/latex] around the line [latex]x=-2.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167794138246\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167794138246\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794138246\">[latex]\\frac{11\\pi }{6}[\/latex] units<sup>3<\/sup><\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/3Rq70sJECwQ?controls=0&amp;start=1099&amp;end=1246&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/2.3VolumesOfRevolutionCylindricalShells1099to1246_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;2.3 Volumes of Revolution: Cylindrical Shells&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Define [latex]R[\/latex] as the region bounded above by the graph of [latex]f(x)=x[\/latex] and below by the graph of [latex]g(x)={x}^{2}[\/latex] over the interval [latex]\\left[0,1\\right].[\/latex] Find the volume of the solid of revolution formed by revolving [latex]R[\/latex] around the [latex]y\\text{-axis}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1167793691567\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1167793691567\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793691567\">[latex]\\frac{\\pi }{6}[\/latex] units<sup>3<\/sup><\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":6,"menu_order":20,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":421,"module-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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/441"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/441\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/421"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/441\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=441"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=441"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=441"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=441"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}