**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.

- \(x^2+y^2 = 4, \; z =-1\)
- \(y< 8\)
- \(0 \leq z \leq 6\)
- \( 1 \leq x^2 + y^2 + z^2 \leq 4\)
- \(x^2 + z^2 \leq 9 \)
- \(x^2 + y^2 + z^2 > 2 z\)
- \(x^2+z^2 =16\)
- \(z^2 =1\)
- \(x^2 + y^2 + z^2 \leq 3\)
- \(x \geq -3 \)
- \(x^2 + y^2 + z^2 > 2 z, \; x \geq 0\)
- \(x=z\)

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:

- A shell centered at \((1,-2,1)\) with thickness 1, and inner radius 1.
- A sphere of radius 2 centered at the origin
- A solid cylinder with \(x\)-axis going through its center with radius 2
- A solid cylinder with \(y\)-axis going through its center radius 1
- 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.