{"id":931,"date":"2025-10-14T16:20:25","date_gmt":"2025-10-14T16:20:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=931"},"modified":"2025-11-13T14:55:37","modified_gmt":"2025-11-13T14:55:37","slug":"alternative-coordinate-systems-and-surfaces-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/alternative-coordinate-systems-and-surfaces-learn-it-3\/","title":{"raw":"Alternative Coordinate Systems and Surfaces: Learn It 3","rendered":"Alternative Coordinate Systems and Surfaces: Learn It 3"},"content":{"raw":"

Cylinders and Surfaces in 3D<\/h2>\r\n

We've explored vectors, lines, planes, and spheres in three-dimensional space. Now we'll investigate a broader variety of surfaces that can be graphed in 3D coordinate systems, starting with cylinders.<\/p>\r\n\r\n

Identifying Cylinders<\/h3>\r\nWhen most people hear \"cylinder,\" they think of a hollow pipe or soda can. But the mathematical definition is much broader\u2014a rectangular heating duct is a cylinder, as is a rolled-up yoga mat with spiral cross-section.\r\n

Consider the equation [latex]x^2 + y^2 = 9[\/latex]. In 2D, this describes a circle centered at the origin with radius 3. In 3D, this same equation represents a surface. Imagine stacking copies of this circle on top of each other, all centered on the [latex]z[\/latex]-axis. This creates a hollow tube extending infinitely in both directions along the [latex]z[\/latex]-axis.<\/p>\r\n\r\n[caption id=\"attachment_5201\" align=\"aligncenter\" width=\"350\"]\"This 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

cylindrical surface<\/h3>\r\nA set of parallel lines (called rulings<\/strong>) passing through a given curve forms a cylindrical surface, or cylinder<\/strong>. The rulings can be parallel to any line, and the original curve can have any shape.\r\n\r\n<\/section>This definition shows that cylinders don't need circular cross-sections. Any curve can form a cylinder, and the parallel lines can point in any direction.\r\n\r\n[caption id=\"attachment_5204\" align=\"aligncenter\" width=\"434\"]\"This 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

Identifying Cylinders<\/strong>: An equation describes a cylinder if one variable is completely missing. The missing variable corresponds to the direction of the rulings:<\/p>\r\n\r\n

    \r\n \t
  • Missing [latex]z[\/latex]: rulings parallel to [latex]z[\/latex]-axis<\/li>\r\n \t
  • Missing [latex]y[\/latex]: rulings parallel to [latex]y[\/latex]-axis<\/li>\r\n \t
  • Missing [latex]x[\/latex]: rulings parallel to [latex]x[\/latex]-axis<\/li>\r\n<\/ul>\r\n<\/section>
    \r\n
    \r\n
    \r\n

    Sketch the graphs of the following cylindrical surfaces.<\/p>\r\n\r\n

      \r\n \t
    1. [latex]x^{2}+z^{2}=25[\/latex]<\/li>\r\n \t
    2. [latex]z=2x^{2}-y[\/latex]<\/li>\r\n \t
    3. [latex]y=\\sin x[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
      \r\n
      <\/div>\r\n<\/div>\r\n[reveal-answer q=\"475862987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"475862987\"]\r\n
        \r\n \t
      1. 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\"]\"This 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
      2. 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\"]\"This Figure 4. The graph of the equation [latex]z=2x^{2}-y[\/latex].[\/caption]<\/li>\r\n \t
      3. 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\"]\"This 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<\/section>\r\n

        Traces: Cross-Sections of Surfaces<\/h3>\r\n

        When sketching surfaces, it helps to examine where the surface intersects planes parallel to the coordinate planes. These intersections are called traces<\/strong>.<\/p>\r\n\r\n

        \r\n

        traces<\/h3>\r\nThe traces of a surface are the cross-sections created when the surface intersects a plane parallel to one of the coordinate planes ([latex]xy[\/latex], [latex]xz[\/latex], or [latex]yz[\/latex] planes).\r\n\r\n<\/section>[caption id=\"attachment_5212\" align=\"aligncenter\" width=\"889\"]\"This 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]
        \r\n

        For cylindrical surfaces, only one set of traces provides useful information: The trace perpendicular to the rulings shows the shape of the generating curve. Traces parallel to the rulings just show parallel lines.<\/p>\r\n

        For example, with [latex]z = \\sin x[\/latex]:<\/p>\r\n\r\n

          \r\n \t
        • [latex]xz[\/latex]-plane trace<\/strong> ([latex]y = 0[\/latex]): Shows the sine wave that generates the cylinder<\/li>\r\n \t
        • [latex]xy[\/latex]-plane trace<\/strong> ([latex]z = 0[\/latex]): Shows parallel lines (not very informative)<\/li>\r\n \t
        • [latex]yz[\/latex]-plane trace<\/strong> ([latex]x = 0[\/latex]): Shows a single line (also not informative)<\/li>\r\n<\/ul>\r\n
          \r\n
          \r\n

          Traces become an essential tool as we explore more complex surfaces beyond cylinders, where the geometry isn't as straightforward as parallel lines through a curve.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

          \u00a0Cylindrical surfaces are the simplest type of 3D surface\u2014they're formed by parallel lines. More complex surfaces require examining multiple traces to understand their shape fully.<\/section><\/section>","rendered":"

          Cylinders and Surfaces in 3D<\/h2>\n

          We’ve explored vectors, lines, planes, and spheres in three-dimensional space. Now we’ll investigate a broader variety of surfaces that can be graphed in 3D coordinate systems, starting with cylinders.<\/p>\n

          Identifying Cylinders<\/h3>\n

          When most people hear “cylinder,” they think of a hollow pipe or soda can. But the mathematical definition is much broader\u2014a rectangular heating duct is a cylinder, as is a rolled-up yoga mat with spiral cross-section.<\/p>\n

          Consider the equation [latex]x^2 + y^2 = 9[\/latex]. In 2D, this describes a circle centered at the origin with radius 3. In 3D, this same equation represents a surface. Imagine stacking copies of this circle on top of each other, all centered on the [latex]z[\/latex]-axis. This creates a hollow tube extending infinitely in both directions along the [latex]z[\/latex]-axis.<\/p>\n

          \"This
          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.<\/figcaption><\/figure>\n
          \n

          cylindrical surface<\/h3>\n

          A set of parallel lines (called rulings<\/strong>) passing through a given curve forms a cylindrical surface, or cylinder<\/strong>. The rulings can be parallel to any line, and the original curve can have any shape.<\/p>\n<\/section>\n

          This definition shows that cylinders don’t need circular cross-sections. Any curve can form a cylinder, and the parallel lines can point in any direction.<\/p>\n

          \"This
          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.<\/figcaption><\/figure>\n
          \n

          Identifying Cylinders<\/strong>: An equation describes a cylinder if one variable is completely missing. The missing variable corresponds to the direction of the rulings:<\/p>\n

            \n
          • Missing [latex]z[\/latex]: rulings parallel to [latex]z[\/latex]-axis<\/li>\n
          • Missing [latex]y[\/latex]: rulings parallel to [latex]y[\/latex]-axis<\/li>\n
          • Missing [latex]x[\/latex]: rulings parallel to [latex]x[\/latex]-axis<\/li>\n<\/ul>\n<\/section>\n
            \n
            \n
            \n

            Sketch the graphs of the following cylindrical surfaces.<\/p>\n

              \n
            1. [latex]x^{2}+z^{2}=25[\/latex]<\/li>\n
            2. [latex]z=2x^{2}-y[\/latex]<\/li>\n
            3. [latex]y=\\sin x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
              \n
              <\/div>\n<\/div>\n