Answer
We derive formula (8) 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_{\phi = {\phi _1}}^{{\phi _2}} \mathop \smallint \limits_{\rho = {\rho _1}\left( {\theta ,\phi } \right)}^{{\rho _2}\left( {\theta ,\phi } \right)} f\left( {\rho \sin \phi \cos \theta ,\rho \sin \phi \sin \theta ,\rho \cos \phi } \right){\rho ^2}\sin \phi {\rm{d}}\rho {\rm{d}}\phi {\rm{d}}\theta $
Work Step by Step
In spherical coordinates, the mapping from the region ${{\cal W}_0}$ in $\left( {\rho ,\phi ,\theta } \right)$-space to the region ${\cal W}$ in $\left( {x,y,z} \right)$-space is given by
$x = \rho \sin \phi \cos \theta $, ${\ \ \ }$ $y = \rho \sin \phi \sin \theta $, ${\ \ \ }$ $z = \rho \cos \phi $
Let the region description of ${{\cal W}_0}$ be given by
${{\cal W}_0} = \left\{ {\left( {\rho ,\phi ,\theta } \right)|{\rho _1}\left( {\theta ,\phi } \right) \le \rho \le {\rho _2}\left( {\theta ,\phi } \right),{\phi _1} \le \phi \le {\phi _2},{\theta _1} \le \theta \le {\theta _2}} \right\}$
Write the mapping:
$G\left( {\rho ,\phi ,\theta } \right) = \left( {\rho \sin \phi \cos \theta ,\rho \sin \phi \sin \theta ,\rho \cos \phi } \right)$
Evaluate the Jacobian of $G$:
${\rm{Jac}}\left( G \right) = \left| {\begin{array}{*{20}{c}}
{\frac{{\partial x}}{{\partial \rho }}}&{\frac{{\partial x}}{{\partial \phi }}}&{\frac{{\partial x}}{{\partial \theta }}}\\
{\frac{{\partial y}}{{\partial \rho }}}&{\frac{{\partial y}}{{\partial \phi }}}&{\frac{{\partial y}}{{\partial \theta }}}\\
{\frac{{\partial z}}{{\partial \rho }}}&{\frac{{\partial z}}{{\partial \phi }}}&{\frac{{\partial z}}{{\partial \theta }}}
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
{\sin \phi \cos \theta }&{\rho \cos \phi \cos \theta }&{ - \rho \sin \phi \sin \theta }\\
{\sin \phi \sin \theta }&{\rho \cos \phi \sin \theta }&{\rho \sin \phi \cos \theta }\\
{\cos \phi }&{ - \rho \sin \phi }&0
\end{array}} \right|$
$ = \sin \phi \cos \theta \left( {{\rho ^2}{{\sin }^2}\phi \cos \theta } \right)$
${\ \ }$ $ - \rho \cos \phi \cos \theta \left( { - \rho \cos \phi \sin \phi \cos \theta } \right)$
${\ \ }$ $ - \rho \sin \phi \sin \theta \left( { - \rho {{\sin }^2}\phi \sin \theta - \rho {{\cos }^2}\phi \sin \theta } \right)$
$ = {\rho ^2}{\sin ^3}\phi {\cos ^2}\theta + {\rho ^2}{\cos ^2}\phi \sin \phi {\cos ^2}\theta + \rho \sin \phi \sin \theta \left( {\rho \sin \theta } \right)$
$ = {\rho ^2}{\sin ^3}\phi {\cos ^2}\theta + {\rho ^2}{\cos ^2}\phi \sin \phi {\cos ^2}\theta + {\rho ^2}\sin \phi {\sin ^2}\theta $
$ = \sin \phi {\cos ^2}\theta \left( {{\rho ^2}{{\sin }^2}\phi + {\rho ^2}{{\cos }^2}\phi } \right) + {\rho ^2}\sin \phi {\sin ^2}\theta $
$ = {\rho ^2}\sin \phi {\cos ^2}\theta + {\rho ^2}\sin \phi$
$ = {\rho ^2}\sin \phi $
So, ${\rm{Jac}}\left( G \right) = {\rho ^2}\sin \phi $.
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( {\rho ,\phi ,\theta } \right),y\left( {\rho ,\phi ,\theta } \right),z\left( {\rho ,\phi ,\theta } \right)} \right)\left| {Jac\left( G \right)} \right|{\rm{d}}\rho {\rm{d}}\phi {\rm{d}}\theta $
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y{\rm{d}}z$
$ = \mathop \smallint \limits_{\theta = {\theta _1}}^{{\theta _2}} \mathop \smallint \limits_{\phi = {\phi _1}}^{{\phi _2}} \mathop \smallint \limits_{\rho = {\rho _1}\left( {\theta ,\phi } \right)}^{{\rho _2}\left( {\theta ,\phi } \right)} f\left( {\rho \sin \phi \cos \theta ,\rho \sin \phi \sin \theta ,\rho \cos \phi } \right){\rho ^2}\sin \phi {\rm{d}}\rho {\rm{d}}\phi {\rm{d}}\theta $
The last integral is formula (8) in Section 16.4.
