Answer
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}V $
$= \mathop \smallint \limits_{z = 0}^1 \mathop \smallint \limits_{y = - z}^z \mathop \smallint \limits_{x = - \sqrt {{z^2} - {y^2}} }^{\sqrt {{z^2} - {y^2}} } f\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y{\rm{d}}z$
Work Step by Step
Recall from Exercise 29, we have the solid region given by
${\cal W} = \left\{ {\left( {x,y,z} \right):\sqrt {{x^2} + {y^2}} \le z \le 1} \right\}$
To express $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}V$ as an iterated integral in the order $dxdydz$, we project ${\cal W}$ onto the $yz$-plane to obtain the domain ${\cal T}$. On the $yz$-plane, that is, $x=0$, we obtain $z = \pm y$. Thus, the domain ${\cal T}$ is a triangle bounded left by $z=-y$ and bounded right by $z=y$ (please see the figure attached). We choose to describe ${\cal T}$ as a horizontally simple region, thus
${\cal T} = \left\{ {\left( {x,y} \right)|0 \le z \le 1, - z \le y \le z} \right\}$
Referring to Figure 16 and the figure attached, we see that ${\cal W}$ is located on both positive and negative $x$-axis. Using $z = \sqrt {{x^2} + {y^2}} $, we obtain $x = \pm \sqrt {{z^2} - {y^2}} $. Thus, we can describe it as a $x$-simple region bounded below by $x = - \sqrt {{z^2} - {y^2}} $ and bounded above by $x = \sqrt {{z^2} - {y^2}} $. Thus, the region description of ${\cal W}$:
${\cal W} = \left\{ {\left( {x,y,z} \right)|0 \le z \le 1, - z \le y \le z, - \sqrt {{z^2} - {y^2}} \le x \le \sqrt {{z^2} - {y^2}} } \right\}$
The triple integral is equal to the iterated integral:
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}V $
$= \mathop \smallint \limits_{z = 0}^1 \mathop \smallint \limits_{y = - z}^z \mathop \smallint \limits_{x = - \sqrt {{z^2} - {y^2}} }^{\sqrt {{z^2} - {y^2}} } f\left( {x,y,z} \right){\rm{d}}x{\rm{d}}y{\rm{d}}z$
