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Surfaces in Space: Definition of a Cylinder

Learn about the definition of a cylinder in space, including its generating curve and rulings. Explore examples and equations of quadric surfaces. Discover how to sketch quadric surfaces and identify their regions.

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Surfaces in Space: Definition of a Cylinder

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  1. 11.6 Surfaces in Space

  2. Definition of a Cylinder • Let C be a curve in a plane and let L be a line not in a parallel plane. The set of all lines parallel to L and intersecting C is called a cylinder. C is called the generating curve (or directrix), and the parallel lines are called rulings. • Note: If one of the variables is missing from the equation of a cylinder, its rulings are parallel to the coordinate axis of the missing variable.

  3. Examples:

  4. Quadric Surfaces • The equation of a quadric surface in space is a second-degree equation of the form • There are six basic types of quadric surfaces:

  5. 1) Ellipsoid • Standard Form

  6. 2) Hyperboloids of One Sheet • Standard Equation:

  7. 3) Hyperboloid of Two Sheets • Standard equation

  8. 4) Elliptic Cone • Standard Equation:

  9. 5) Elliptic Paraboloid • Standard Equation:

  10. 6) Hyperbolic Paraboloid • Standard Equation:

  11. To Sketch a Quadric Surface • Write the surface in standard form. • Determine the traces in the coordinate planes by setting each variable =0 For example: To get the trace in the xy-plane, set z=0. To get the trace in the xz-plane, set y=0, etc. • If needed, find the traces in planes that are parallel to coordinate planes by holding a variable constant.

  12. Examples: Identify and Sketch: 1) 2)

  13. #42 • Sketch the region bounded by the graphs of the equations. X=0, y=0, z=0

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