Units

# 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:

- 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.
- 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:

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

Use the graph to find the answers to these questions.