Lesson 4.2: Surfaces

In this lesson, you will learn about cylinders, quadric surfaces, and traces.

View all of the instructional videos, read the provided readings, and complete the practice exercises/activities. These will help you master the objectives for this lesson. Pay attention to additional comments provided. Try to visualize the problems. The reason that we analyze traces is to help us to visualize the shape of the surface. It can be extremely helpful if you review conic sections: parabolas, ellipses (including circles), and hyperbolas.

Lesson Objectives

Upon completion of the lesson 4.2, you will be able to:

• Identify and analyze cylinders

• Identify and analyze quadric surfaces

• Match graphs with equations

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

1. Cylindrical Surfaces [8:18]

   1. Source: bullcleo1 from YouTube

1. Introduction to Quadric Surfaces [7:48]

   1. Source: bullcleo1 from YouTube

• 1. Usually we need to analyze the traces more generally, that is, we need to analyze traces that are in the planes parallel to the coordinate planes, in order to visualize and determine the properties of a given surface. It is possible that a surface does not intersect one of the coordinate planes: in this case, there will be no trace in that coordinate plane. Watch the next six videos to obtain a basic understanding of the six quadric surfaces that we are going to study and how to obtain some particular traces. The comments for the videos below, will show you how to generally analyze a given surface. Use the examples that I showed as a reference when working on your "Show Your Work" assignment.

1. Graphing Quadric Surfaces: The Ellipsoid [5:55]

   1. Source: bullcleo1 from YouTube

1. Graphing Quadric Surfaces: The Elliptical Paraboloid [7:46]

   1. Source: bullcleo1 from YouTube

1. Graphing Quadric Surfaces: The Hyperboloid of One Sheet [6:31]

   1. Source: bullcleo1 from YouTube

1. Graphing Quadric Surfaces: The Hyperboloid of Two Sheets [8:37]

   1. Source: bullcleo1 from YouTube

1. Graphing Quadric Surfaces: The Elliptical Cone [7:20]

   1. Source: bullcleo1 from YouTube

1. Graphing Quadric Surfaces: The Hyperbolic Paraboloid [7:46]

   1. Source: bullcleo1 from YouTube

1. Space Curves Represented by the Intersection of Surfaces [9:05]

   1. Source: bullcleo1 from YouTube

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.

1. Quadric Surfaces

2. Paul's Online Math Notes

3. Calculus III | Graphs of Quadric Surfaces

4. 10.6 Surfaces in Space

   1. (cylindrical surfaces, quadric surfaces, an application)

5. Cylinders and Quadric Surfaces

• 1. Identify and sketch the surface 4x2 – y2 + 2z2 + 4 = 0.

  2. Cylinders (right circular cylinder, other right cylinders, a "not right" cylinders, and traces of cylinders)

  3. Blue Ridge Community College

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

1. Interactive Gallery of Quadric Surfaces

   1. Use the link below to explore more features of the quadric surfaces.

2. 12.6 Cylinders and Quadric Surfaces

• 1. Click on "12 Vectors and the Geometry of Space" then choose "12.6 Cylinders and Quadric Surfaces"

Additional Resources

Below are additional resources that help reinforce the content for this module.

1. 1.6 Quadric Surfaces

Homework Assignment

Do Homework 4.2 on MyMathLab.

Show Your Work Assignment

Show Your Work assignment will be given via Assignment Tool on Laulima. Please check your email for Laulima announcement.

Lesson 4.3: Graphs and Level Surfaces -->