# 12.1 | Functions of Two Variables

Distance Formula in Three Dimensions: The distance $$\left| P_1 P_2 \right|$$ between the points $$P_1(x_1,y_1,z_1)$$ and $$P_2(x_2, y_2, z_2)$$ is
$$\left| P_2 P_2\right| = \sqrt{ (x_2 – x_1)^2 + (y_2-y_1)^2 + (z_2 – z_1)^2 }$$
Equation of a Sphere: An equation of a sphere with center $$C(h,k,l)$$ and radius $$r$$ is
$$(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2$$
In particular, if the center is the origin $$O$$, then an equation of the sphere is $$x^2+y^2+z^2 = r^2$$.

Match the graphs on the left to the list of equalities or inequalities below.

1. $$x^2+y^2 = 4, \; z =-1$$
2. $$y< 8$$
3. $$0 \leq z \leq 6$$
4. $$1 \leq x^2 + y^2 + z^2 \leq 4$$
5. $$x^2 + z^2 \leq 9$$
6. $$x^2 + y^2 + z^2 > 2 z$$
7. $$x^2+z^2 =16$$
8. $$z^2 =1$$
9. $$x^2 + y^2 + z^2 \leq 3$$
10. $$x \geq -3$$
11. $$x^2 + y^2 + z^2 > 2 z, \; x \geq 0$$
12. $$x=z$$

Use the demonstration on the left to find values of $$a,b,c,h,k,l,e,$$ and $$d$$ such that the resulting region is:

1. A shell centered at $$(1,-2,1)$$ with thickness 1, and inner radius 1.
2. A sphere of radius 2 centered at the origin
3. A solid cylinder with $$x$$-axis going through its center with radius 2
4. A solid cylinder with $$y$$-axis going through its center radius 1
5. A cylinderical shell with thickness 1 and outer radius 2 with center along the line through the point $$(0,0,1)$$ and in the direction $$\langle 0,1,0 \rangle$$

Write down the set of inequalities or equalities that correspond to that region.