Unit 10: 3D Solid Bounded by Two Surfaces

The Concept 

The graphs of functions of two variables [latex]z=f(x, y)[/latex] are examples of surfaces in 3D. More generally, a set of points [latex](x,y,z)[/latex] that satisfy an equation relating all three variables is often a surface. A simple example is the unit sphere, the set of points that satisfy the equation [latex]x^2+y^2+z^2=1[/latex].

One special class of equations is a set of equations that involve one or more [latex]x^2, y^2, z^2, xy, xz[/latex], and [latex]yz[/latex]. The graphs of these equations are surfaces known as quadric surfaces. There are six different quadric surfaces: the ellipsoid, the elliptic paraboloid, the hyperbolic paraboloid, the double cone and hyperboloids of one sheet and two sheets. Quadric surfaces are natural 3D extensions of the so-called conics (ellipses, parabolas and hyperbolas), and they provide examples of fairly nice surfaces to use as examples in multivariate calculus.

The Plot

Now, you should engage with the 3D plot below to understand 3D solids bounded by two surfaces[1]. Follow the steps below to apply changes to the plot and observe the effects:

  1. Fill in function 1 (i.e., [latex]f(x,y,z)[/latex]) and function 2 (i.e., [latex]g(x,y,z)[/latex]) with your desired quadric surfaces.
  2. The graph depicted on the right shows their intersection.

Self-Checking Questions

Check your understanding by solving the following questions[2]:

Plot the given quadric surface and specify the name of said quadric surface:

  1. [latex]x^2/4 + y^2/9 - z^2/12 = 1[/latex]
  2. [latex]z^2 = 4x^2 + 3y^2[/latex]

Use the graph to find the answers to these questions.


  1. Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.
  2. Gilbert Strang, Edwin “Jed” Herman,  OpenStax, Calculus Volume 3,  Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).


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3D Interactive Plots for Multivariate Calculus Copyright © 2022 by Dr. Na Yu, Ryerson University is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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