Units

# Unit 10: 3D Solid Bounded by Two Surfaces

## The Concept

The graphs of functions of two variables $z=f(x, y)$ are examples of surfaces in 3D. More generally, a set of points $(x,y,z)$ 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 $x^2+y^2+z^2=1$.

One special class of equations is a set of equations that involve one or more $x^2, y^2, z^2, xy, xz$, and $yz$. 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., $f(x,y,z)$) and function 2 (i.e., $g(x,y,z)$) 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. $x^2/4 + y^2/9 - z^2/12 = 1$
2. $z^2 = 4x^2 + 3y^2$

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).

## License

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.