Answer
The inequalities in cylindrical coordinates:
$ - \sqrt 3 \le z \le \sqrt 3 $ ${\ \ }$ and ${\ \ }$ $1 \le r \le \sqrt {4 - {z^2}} $.
Work Step by Step
In cylindrical coordinates.
1. the entire apple:
The apple is modeled by the equation: ${x^2} + {y^2} + {z^2} \le 4$. This is a ball of radius $2$.
Since ${r^2} = {x^2} + {y^2}$, converting to cylindrical coordinates we get ${r^2} + {z^2} \le 4$.
2. the core:
The core of the apple is a vertical cylinder of radius $1$. So, $r \le 1$.
3. The apple without the core:
From point 1 and point 2, we get ${r^2} + {z^2} \le 4$ and $r \ge 1$ (excluding the core)
$r \le \sqrt {4 - {z^2}} $ ${\ \ }$ and ${\ \ }$ $r \ge 1$
This is equivalent to $1 \le r \le \sqrt {4 - {z^2}} $.
Notice that the inequality implicitly implies that $4 - {z^2} \ge 1$. So,
${z^2} \le 3$ ${\ \ }$ or ${\ \ }$ $ - \sqrt 3 \le z \le \sqrt 3 $
Therefore, the inequalities in cylindrical coordinates:
$ - \sqrt 3 \le z \le \sqrt 3 $ ${\ \ }$ and ${\ \ }$ $1 \le r \le \sqrt {4 - {z^2}} $.
