{"id":5461,"date":"2022-06-02T18:33:50","date_gmt":"2022-06-02T18:33:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=5461"},"modified":"2022-10-26T02:55:11","modified_gmt":"2022-10-26T02:55:11","slug":"identifying-cylinders","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/identifying-cylinders\/","title":{"raw":"Identifying Cylinders","rendered":"Identifying Cylinders"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul class=\"os-abstract\">\r\n \t<li><span class=\"os-abstract-content\">Identify a cylinder as a type of three-dimensional surface.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1163723183468\">The first surface we\u2019ll examine is the cylinder. Although most people immediately think of a hollow pipe or a soda straw when they hear the word\u00a0<em data-effect=\"italics\">cylinder<\/em>, here we use the broad mathematical meaning of the term. As we have seen, cylindrical surfaces don\u2019t have to be circular. A rectangular heating duct is a cylinder, as is a rolled-up yoga mat, the cross-section of which is a spiral shape.<\/p>\r\n<p id=\"fs-id1163723994234\">In the two-dimensional coordinate plane, the equation [latex]x^{2}+y^{2}=9[\/latex] describes a circle centered at the origin with radius 3. In three-dimensional space, this same equation represents a surface. Imagine copies of a circle stacked on top of each other centered on the [latex]z[\/latex]-axis (Figure 1), forming a hollow tube. We can then construct a cylinder from the set of lines parallel to the [latex]z[\/latex]-axis passing through circle [latex]x^{2}+y^{2}=9[\/latex] in the [latex]xy[\/latex]-plane, as shown in the figure. In this way, any curve in one of the coordinate planes can be extended to become a surface.<\/p>\r\n\r\n[caption id=\"attachment_5201\" align=\"aligncenter\" width=\"350\"]<img class=\"size-full wp-image-5201\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31202400\/2.75.jpg\" alt=\"This figure a 3-dimensional coordinate system. It has a right circular center with the z-axis through the center. The cylinder also has points labeled on the x and y axis at (3, 0, 0) and (0, 3, 0).\" width=\"350\" height=\"400\" \/> Figure 1. In three-dimensional space, the graph of equation [latex]x^{2}+y^{2}=9[\/latex] is a cylinder with radius [latex]3[\/latex] centered on the [latex]z[\/latex]-axis. It continues indefinitely in the positive and negative directions.[\/caption]\r\n<div class=\"textbox shaded\"><header>\r\n<h3 class=\"os-title\" style=\"text-align: center;\" data-type=\"title\"><span id=\"3\" class=\"os-title-label\" data-type=\"\">DEFINITION<\/span><\/h3>\r\n\r\n<hr \/>\r\n\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1163724084014\">A set of lines parallel to a given line passing through a given curve is known as a cylindrical surface, or\u00a0<strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term96\" data-type=\"term\">cylinder<\/span>.<\/strong> The parallel lines are called\u00a0<strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term97\" data-type=\"term\">rulings<\/span>.<\/strong><\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\nFrom this definition, we can see that we still have a cylinder in three-dimensional space, even if the curve is not a circle. Any curve can form a cylinder, and the rulings that compose the cylinder may be parallel to any given line (Figure 2).\r\n\r\n[caption id=\"attachment_5204\" align=\"aligncenter\" width=\"434\"]<img class=\"size-full wp-image-5204\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31202529\/2.76.jpg\" alt=\"This figure has a 3-dimensional surface that begins on the y-axis and curves upward. There is also the x and z axes labeled.\" width=\"434\" height=\"441\" \/> Figure 2. In three-dimensional space, the graph of equation [latex]z=x^{3}[\/latex] is a cylinder, or a cylindrical surface with rulings parallel to the [latex]y[\/latex]-axis.[\/caption]\r\n<div class=\"textbox exercises\">\r\n<h3>Example: graphing cylindrical surfaces<\/h3>\r\n<div id=\"fs-id1163723242452\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1163723656797\">Sketch the graphs of the following cylindrical surfaces.<\/p>\r\n\r\n<ol id=\"fs-id1163724075529\" type=\"a\">\r\n \t<li>[latex]x^{2}+z^{2}=25[\/latex]<\/li>\r\n \t<li>[latex]z=2x^{2}-y[\/latex]<\/li>\r\n \t<li>[latex]y=\\sin x[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1163723281261\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\r\n<div class=\"ui-toggle-wrapper\"><\/div>\r\n<\/div>\r\n[reveal-answer q=\"475862987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"475862987\"]\r\n<ol id=\"fs-id1163723864671\" type=\"a\">\r\n \t<li>The variable [latex]y[\/latex] can take on any value without limit. Therefore, the lines ruling this surface are parallel to the [latex]y[\/latex]-axis. The intersection of this surface with the [latex]xz[\/latex]-plane forms a circle centered at the origin with radius 5 (see the following figure).\r\n[caption id=\"attachment_5206\" align=\"aligncenter\" width=\"422\"]<img class=\"size-full wp-image-5206\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31202743\/2.77.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has a right circular cylinder on its side with the y-axis in the center. The cylinder intersects the x-axis at (5, 0, 0). It also has two points of intersection labeled on the z-axis at (0, 0, 5) and (0, 0, -5).\" width=\"422\" height=\"357\" \/> Figure 3. The graph of equation [latex]x^{2}+z^{2}=25[\/latex] is a cylinder with radius [latex]5[\/latex] centered on the [latex]y[\/latex]-axis.[\/caption]<\/li>\r\n \t<li>In this case, the equation contains all three variables\u2013[latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex]\u2013so none of the variables can vary arbitrarily. The easiest way to visualize this surface is to use a computer graphing utility (see the following figure).[caption id=\"attachment_5207\" align=\"aligncenter\" width=\"425\"]<img class=\"size-full wp-image-5207\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31202816\/2.78.jpg\" alt=\"This figure has a surface in the first octant. The cross section of the solid is a parabola.\" width=\"425\" height=\"429\" \/> Figure 4. The graph of the equation [latex]z=2x^{2}-y[\/latex].[\/caption]<\/li>\r\n \t<li>In this equation, the variable [latex]z[\/latex] can take on any value without limit. Therefore, the lines composing this surface are parallel to the [latex]z[\/latex]-axis. The intersection of this surface with the [latex]xy[\/latex]-plane outlines curve [latex]y=\\sin x[\/latex] (see the following figure).\r\n[caption id=\"attachment_5208\" align=\"aligncenter\" width=\"556\"]<img class=\"size-full wp-image-5208\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31202843\/2.79.jpg\" alt=\"This figure is a three dimensional surface. A cross section of the surface parallel to the x y plane would be the sine curve.\" width=\"556\" height=\"584\" \/> Figure 5. The graph of equation [latex]y=\\sin x[\/latex] is formed by a set of lines parallel to the [latex]z[\/latex]-axis passing through curve [latex]y=\\sin x[\/latex] in the [latex]xy[\/latex]-plane.[\/caption]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nSketch or use a graphing tool to view the graph of the cylindrical surface defined by equation [latex]z=y^{2}[\/latex].\r\n\r\n[reveal-answer q=\"378465244\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"378465244\"]\r\n\r\n[caption id=\"attachment_5210\" align=\"aligncenter\" width=\"288\"]<img class=\"size-full wp-image-5210\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203348\/2.52.jpg\" alt=\"This figure is a surface above the x y plane. A cross section of this surface parallel to the y z plane would be a parabola. The surface sits on top of the x y plane.\" width=\"288\" height=\"324\" \/> Figure 6. The cylindrical surface defined by equation [latex]z=y^{2}[\/latex].[\/caption][\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try IT.\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7809453&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=ygBugCYrufQ&amp;video_target=tpm-plugin-mxtub258-ygBugCYrufQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.52_transcript.html\">\u201cCP 2.52\u201d here (opens in new window).<\/a><\/span><\/center>\r\n<div data-type=\"note\">\r\n<div id=\"fs-id1163724036465\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n<div id=\"fs-id1163724074875\" class=\"theorem ui-has-child-title\" data-type=\"note\">\r\n<div id=\"fs-id1163724080388\" class=\"ui-has-child-title\" data-type=\"example\">\r\n<div data-type=\"example\">When sketching surfaces, we have seen that it is useful to sketch the intersection of the surface with a plane parallel to one of the coordinate planes. These curves are called traces. We can see them in the plot of the cylinder in\u00a0Figure 7.<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"example\"><\/div>\r\n<div data-type=\"example\">\r\n<div class=\"textbox shaded\"><header>\r\n<h3 class=\"os-title\" style=\"text-align: center;\" data-type=\"title\"><span id=\"11\" class=\"os-title-label\" data-type=\"\">DEFINITION<\/span><\/h3>\r\n\r\n<hr \/>\r\n\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1163724024018\">The\u00a0<strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term98\" data-type=\"term\">traces<\/span><\/strong>\u00a0of a surface are the cross-sections created when the surface intersects a plane parallel to one of the coordinate planes.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n[caption id=\"attachment_5212\" align=\"aligncenter\" width=\"889\"]<img class=\"size-full wp-image-5212\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203455\/2.80.jpg\" alt=\"This figure has two images. The first image is a surface. A cross section of the surface parallel to the x z plane would be a sine curve. The second image is the sine curve in the x y plane.\" width=\"889\" height=\"499\" \/> Figure 7. (a) This is one view of the graph of equation [latex]z=\\sin x[\/latex]. (b) To find the trace of the graph in the [latex]xz[\/latex]-plane, set [latex]y=0[\/latex]. The trace is simply a two-dimensional sine wave.[\/caption]<section id=\"fs-id1163723958933\" data-depth=\"1\">\r\n<p id=\"fs-id1163723110833\">Traces are useful in sketching cylindrical surfaces. For a cylinder in three dimensions, though, only one set of traces is useful. Notice, in\u00a0Figure 7, that the trace of the graph of [latex]z=\\sin x[\/latex] in the [latex]xz[\/latex]-plane is useful in constructing the graph. The trace in the [latex]xy[\/latex]-plane, though, is just a series of parallel lines, and the trace in the [latex]yz[\/latex]-plane is simply one line.<\/p>\r\n<p id=\"fs-id1163724054768\">Cylindrical surfaces are formed by a set of parallel lines. Not all surfaces in three dimensions are constructed so simply, however. We now explore more complex surfaces, and traces are an important tool in this investigation.<\/p>\r\n\r\n<\/section><section id=\"fs-id1163723956780\" data-depth=\"1\"><\/section><\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul class=\"os-abstract\">\n<li><span class=\"os-abstract-content\">Identify a cylinder as a type of three-dimensional surface.<\/span><\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1163723183468\">The first surface we\u2019ll examine is the cylinder. Although most people immediately think of a hollow pipe or a soda straw when they hear the word\u00a0<em data-effect=\"italics\">cylinder<\/em>, here we use the broad mathematical meaning of the term. As we have seen, cylindrical surfaces don\u2019t have to be circular. A rectangular heating duct is a cylinder, as is a rolled-up yoga mat, the cross-section of which is a spiral shape.<\/p>\n<p id=\"fs-id1163723994234\">In the two-dimensional coordinate plane, the equation [latex]x^{2}+y^{2}=9[\/latex] describes a circle centered at the origin with radius 3. In three-dimensional space, this same equation represents a surface. Imagine copies of a circle stacked on top of each other centered on the [latex]z[\/latex]-axis (Figure 1), forming a hollow tube. We can then construct a cylinder from the set of lines parallel to the [latex]z[\/latex]-axis passing through circle [latex]x^{2}+y^{2}=9[\/latex] in the [latex]xy[\/latex]-plane, as shown in the figure. In this way, any curve in one of the coordinate planes can be extended to become a surface.<\/p>\n<div id=\"attachment_5201\" style=\"width: 360px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5201\" class=\"size-full wp-image-5201\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31202400\/2.75.jpg\" alt=\"This figure a 3-dimensional coordinate system. It has a right circular center with the z-axis through the center. The cylinder also has points labeled on the x and y axis at (3, 0, 0) and (0, 3, 0).\" width=\"350\" height=\"400\" \/><\/p>\n<p id=\"caption-attachment-5201\" class=\"wp-caption-text\">Figure 1. In three-dimensional space, the graph of equation [latex]x^{2}+y^{2}=9[\/latex] is a cylinder with radius [latex]3[\/latex] centered on the [latex]z[\/latex]-axis. It continues indefinitely in the positive and negative directions.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<header>\n<h3 class=\"os-title\" style=\"text-align: center;\" data-type=\"title\"><span id=\"3\" class=\"os-title-label\" data-type=\"\">DEFINITION<\/span><\/h3>\n<hr \/>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1163724084014\">A set of lines parallel to a given line passing through a given curve is known as a cylindrical surface, or\u00a0<strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term96\" data-type=\"term\">cylinder<\/span>.<\/strong> The parallel lines are called\u00a0<strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term97\" data-type=\"term\">rulings<\/span>.<\/strong><\/p>\n<\/div>\n<\/section>\n<\/div>\n<p>From this definition, we can see that we still have a cylinder in three-dimensional space, even if the curve is not a circle. Any curve can form a cylinder, and the rulings that compose the cylinder may be parallel to any given line (Figure 2).<\/p>\n<div id=\"attachment_5204\" style=\"width: 444px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5204\" class=\"size-full wp-image-5204\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31202529\/2.76.jpg\" alt=\"This figure has a 3-dimensional surface that begins on the y-axis and curves upward. There is also the x and z axes labeled.\" width=\"434\" height=\"441\" \/><\/p>\n<p id=\"caption-attachment-5204\" class=\"wp-caption-text\">Figure 2. In three-dimensional space, the graph of equation [latex]z=x^{3}[\/latex] is a cylinder, or a cylindrical surface with rulings parallel to the [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: graphing cylindrical surfaces<\/h3>\n<div id=\"fs-id1163723242452\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1163723656797\">Sketch the graphs of the following cylindrical surfaces.<\/p>\n<ol id=\"fs-id1163724075529\" type=\"a\">\n<li>[latex]x^{2}+z^{2}=25[\/latex]<\/li>\n<li>[latex]z=2x^{2}-y[\/latex]<\/li>\n<li>[latex]y=\\sin x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"fs-id1163723281261\" data-type=\"solution\" aria-label=\"show solution\" aria-expanded=\"false\">\n<div class=\"ui-toggle-wrapper\"><\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q475862987\">Show Solution<\/span><\/p>\n<div id=\"q475862987\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1163723864671\" type=\"a\">\n<li>The variable [latex]y[\/latex] can take on any value without limit. Therefore, the lines ruling this surface are parallel to the [latex]y[\/latex]-axis. The intersection of this surface with the [latex]xz[\/latex]-plane forms a circle centered at the origin with radius 5 (see the following figure).\n<div id=\"attachment_5206\" style=\"width: 432px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5206\" class=\"size-full wp-image-5206\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31202743\/2.77.jpg\" alt=\"This figure is the 3-dimensional coordinate system. It has a right circular cylinder on its side with the y-axis in the center. The cylinder intersects the x-axis at (5, 0, 0). It also has two points of intersection labeled on the z-axis at (0, 0, 5) and (0, 0, -5).\" width=\"422\" height=\"357\" \/><\/p>\n<p id=\"caption-attachment-5206\" class=\"wp-caption-text\">Figure 3. The graph of equation [latex]x^{2}+z^{2}=25[\/latex] is a cylinder with radius [latex]5[\/latex] centered on the [latex]y[\/latex]-axis.<\/p>\n<\/div>\n<\/li>\n<li>In this case, the equation contains all three variables\u2013[latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex]\u2013so none of the variables can vary arbitrarily. The easiest way to visualize this surface is to use a computer graphing utility (see the following figure).\n<div id=\"attachment_5207\" style=\"width: 435px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5207\" class=\"size-full wp-image-5207\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31202816\/2.78.jpg\" alt=\"This figure has a surface in the first octant. The cross section of the solid is a parabola.\" width=\"425\" height=\"429\" \/><\/p>\n<p id=\"caption-attachment-5207\" class=\"wp-caption-text\">Figure 4. The graph of the equation [latex]z=2x^{2}-y[\/latex].<\/p>\n<\/div>\n<\/li>\n<li>In this equation, the variable [latex]z[\/latex] can take on any value without limit. Therefore, the lines composing this surface are parallel to the [latex]z[\/latex]-axis. The intersection of this surface with the [latex]xy[\/latex]-plane outlines curve [latex]y=\\sin x[\/latex] (see the following figure).\n<div id=\"attachment_5208\" style=\"width: 566px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5208\" class=\"size-full wp-image-5208\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31202843\/2.79.jpg\" alt=\"This figure is a three dimensional surface. A cross section of the surface parallel to the x y plane would be the sine curve.\" width=\"556\" height=\"584\" \/><\/p>\n<p id=\"caption-attachment-5208\" class=\"wp-caption-text\">Figure 5. The graph of equation [latex]y=\\sin x[\/latex] is formed by a set of lines parallel to the [latex]z[\/latex]-axis passing through curve [latex]y=\\sin x[\/latex] in the [latex]xy[\/latex]-plane.<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Sketch or use a graphing tool to view the graph of the cylindrical surface defined by equation [latex]z=y^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q378465244\">Show Solution<\/span><\/p>\n<div id=\"q378465244\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"attachment_5210\" style=\"width: 298px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5210\" class=\"size-full wp-image-5210\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203348\/2.52.jpg\" alt=\"This figure is a surface above the x y plane. A cross section of this surface parallel to the y z plane would be a parabola. The surface sits on top of the x y plane.\" width=\"288\" height=\"324\" \/><\/p>\n<p id=\"caption-attachment-5210\" class=\"wp-caption-text\">Figure 6. The cylindrical surface defined by equation [latex]z=y^{2}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try IT.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7809453&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=ygBugCYrufQ&amp;video_target=tpm-plugin-mxtub258-ygBugCYrufQ\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.52_transcript.html\">\u201cCP 2.52\u201d here (opens in new window).<\/a><\/span><\/div>\n<div data-type=\"note\">\n<div id=\"fs-id1163724036465\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<div id=\"fs-id1163724074875\" class=\"theorem ui-has-child-title\" data-type=\"note\">\n<div id=\"fs-id1163724080388\" class=\"ui-has-child-title\" data-type=\"example\">\n<div data-type=\"example\">When sketching surfaces, we have seen that it is useful to sketch the intersection of the surface with a plane parallel to one of the coordinate planes. These curves are called traces. We can see them in the plot of the cylinder in\u00a0Figure 7.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\"><\/div>\n<div data-type=\"example\">\n<div class=\"textbox shaded\">\n<header>\n<h3 class=\"os-title\" style=\"text-align: center;\" data-type=\"title\"><span id=\"11\" class=\"os-title-label\" data-type=\"\">DEFINITION<\/span><\/h3>\n<hr \/>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1163724024018\">The\u00a0<strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term98\" data-type=\"term\">traces<\/span><\/strong>\u00a0of a surface are the cross-sections created when the surface intersects a plane parallel to one of the coordinate planes.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"attachment_5212\" style=\"width: 899px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5212\" class=\"size-full wp-image-5212\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203455\/2.80.jpg\" alt=\"This figure has two images. The first image is a surface. A cross section of the surface parallel to the x z plane would be a sine curve. The second image is the sine curve in the x y plane.\" width=\"889\" height=\"499\" \/><\/p>\n<p id=\"caption-attachment-5212\" class=\"wp-caption-text\">Figure 7. (a) This is one view of the graph of equation [latex]z=\\sin x[\/latex]. (b) To find the trace of the graph in the [latex]xz[\/latex]-plane, set [latex]y=0[\/latex]. The trace is simply a two-dimensional sine wave.<\/p>\n<\/div>\n<section id=\"fs-id1163723958933\" data-depth=\"1\">\n<p id=\"fs-id1163723110833\">Traces are useful in sketching cylindrical surfaces. For a cylinder in three dimensions, though, only one set of traces is useful. Notice, in\u00a0Figure 7, that the trace of the graph of [latex]z=\\sin x[\/latex] in the [latex]xz[\/latex]-plane is useful in constructing the graph. The trace in the [latex]xy[\/latex]-plane, though, is just a series of parallel lines, and the trace in the [latex]yz[\/latex]-plane is simply one line.<\/p>\n<p id=\"fs-id1163724054768\">Cylindrical surfaces are formed by a set of parallel lines. Not all surfaces in three dimensions are constructed so simply, however. We now explore more complex surfaces, and traces are an important tool in this investigation.<\/p>\n<\/section>\n<section id=\"fs-id1163723956780\" data-depth=\"1\"><\/section>\n<\/div>\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-5461\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>CP 2.52. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><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":349141,"menu_order":25,"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\"},{\"type\":\"original\",\"description\":\"CP 2.52\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5461","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/5461","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\/349141"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/5461\/revisions"}],"predecessor-version":[{"id":6440,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/5461\/revisions\/6440"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/20"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/5461\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=5461"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=5461"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=5461"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=5461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}