Answer
We derive formula (5) in Section 16.4:
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y{\rm{d}}z$
$ = \mathop \smallint \limits_{\theta = {\theta _1}}^{{\theta _2}} \mathop \smallint \limits_{r = {r_1}\left( \theta \right)}^{{r_2}\left( \theta \right)} \mathop \smallint \limits_{z = {z_1}\left( {r,\theta } \right)}^{{z_2}\left( {r,\theta } \right)} f\left( {r\cos \theta ,r\sin \theta ,z} \right)r{\rm{d}}z{\rm{d}}r{\rm{d}}\theta $
Work Step by Step
In cylindrical coordinates, the mapping from the region ${{\cal W}_0}$ in $\left( {r,\theta ,z} \right)$-space to the region ${\cal W}$ in $\left( {x,y,z} \right)$-space is given by
$x = r\cos \theta $, ${\ \ \ \ }$ $y = r\sin \theta $, ${\ \ \ \ }$ $z=z$
Let the region description of ${{\cal W}_0}$ be given by
${{\cal W}_0} = \left\{ {\left( {r,\theta ,z} \right)|{r_1}\left( \theta \right) \le r \le {r_2}\left( \theta \right),{\theta _1} \le \theta \le {\theta _2},{z_1}\left( {r,\theta } \right) \le z \le {z_2}\left( {r,\theta } \right)} \right\}$
Write the mapping:
$G\left( {r,\theta ,z} \right) = \left( {r\cos \theta ,r\sin \theta ,z} \right)$
Evaluate the Jacobian of $G$:
${\rm{Jac}}\left( G \right) = \left| {\begin{array}{*{20}{c}}
{\frac{{\partial x}}{{\partial r}}}&{\frac{{\partial x}}{{\partial \theta }}}&{\frac{{\partial x}}{{\partial z}}}\\
{\frac{{\partial y}}{{\partial r}}}&{\frac{{\partial y}}{{\partial \theta }}}&{\frac{{\partial y}}{{\partial z}}}\\
{\frac{{\partial z}}{{\partial r}}}&{\frac{{\partial z}}{{\partial \theta }}}&{\frac{{\partial z}}{{\partial z}}}
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
{\cos \theta }&{ - r\sin \theta }&0\\
{\sin \theta }&{r\cos \theta }&0\\
0&0&1
\end{array}} \right|$
$ = \cos \theta \left( {r\cos \theta } \right) - \left( { - r\sin \theta } \right)\sin \theta = r{\cos ^2}\theta + r{\sin ^2}\theta $
So, ${\rm{Jac}}\left( G \right) = r$.
Using the general Change of Variables Formula, Eq. (16), we get
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y{\rm{d}}z$
$ = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{{{\cal W}_0}}^{} f\left( {x\left( {r,\theta ,z} \right),y\left( {r,\theta ,z} \right),z\left( {r,\theta ,z} \right)} \right)\left| {Jac\left( G \right)} \right|{\rm{d}}r{\rm{d}}\theta {\rm{d}}z$
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y{\rm{d}}z$
$ = \mathop \smallint \limits_{\theta = {\theta _1}}^{{\theta _2}} \mathop \smallint \limits_{r = {r_1}\left( \theta \right)}^{{r_2}\left( \theta \right)} \mathop \smallint \limits_{z = {z_1}\left( {r,\theta } \right)}^{{z_2}\left( {r,\theta } \right)} f\left( {r\cos \theta ,r\sin \theta ,z} \right)r{\rm{d}}z{\rm{d}}r{\rm{d}}\theta $
The last integral is formula (5) in Section 16.4.