Answer
The volume of ${\cal W}$:
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} 1{\rm{d}}V = \mathop \smallint \limits_{x = - 1}^1 \mathop \smallint \limits_{y = {x^2}}^1 \mathop \smallint \limits_{z = 0}^{1 - {y^2}} 1{\rm{d}}z{\rm{d}}y{\rm{d}}x = \frac{{16}}{{21}}$
Work Step by Step
We have the region ${\cal W}$ bounded by $z = 1 - {y^2}$, $y = {x^2}$, and the plane $z=0$.
Since ${\cal W}$ is bounded by $y = {x^2}$, $y$ is non-negative. Since ${\cal W}$ is also bounded by $z=0$, using $z = 1 - {y^2}$, we obtain $y=1$ (please see the figure attached).
To obtain a triple integral in the order $dzdydx$, we project ${\cal W}$ onto the $xy$-plane and define the domain ${\cal D}$ obtained by
${\cal D} = \left\{ {\left( {x,y} \right)| - 1 \le x \le 1,{x^2} \le y \le 1} \right\}$
We notice that ${\cal W}$ is $z$-simple region bounded below by $z=0$ and bounded above by $z = 1 - {y^2}$. Thus, the region description of ${\cal W}$ is
${\cal W} = \left\{ {\left( {x,y,z} \right)| - 1 \le x \le 1,{x^2} \le y \le 1,0 \le z \le 1 - {y^2}} \right\}$
The volume of ${\cal W}$ as a triple integral in the order $dzdydx$ is
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} 1{\rm{d}}V = \mathop \smallint \limits_{x = - 1}^1 \mathop \smallint \limits_{y = {x^2}}^1 \mathop \smallint \limits_{z = 0}^{1 - {y^2}} 1{\rm{d}}z{\rm{d}}y{\rm{d}}x$
$ = \mathop \smallint \limits_{x = - 1}^1 \mathop \smallint \limits_{y = {x^2}}^1 \left( {z|_0^{1 - {y^2}}} \right){\rm{d}}y{\rm{d}}x$
$ = \mathop \smallint \limits_{x = - 1}^1 \mathop \smallint \limits_{y = {x^2}}^1 \left( {1 - {y^2}} \right){\rm{d}}y{\rm{d}}x$
$ = \mathop \smallint \limits_{x = - 1}^1 \left( {\left( {y - \frac{1}{3}{y^3}} \right)|_{{x^2}}^1} \right){\rm{d}}x$
$ = \mathop \smallint \limits_{x = - 1}^1 \left( {1 - \frac{1}{3} - {x^2} + \frac{1}{3}{x^6}} \right){\rm{d}}x$
$ = \left( {\frac{2}{3}x - \frac{1}{3}{x^3} + \frac{1}{{21}}{x^7}} \right)|_{ - 1}^1$
$ = \frac{2}{3} - \frac{1}{3} + \frac{1}{{21}} + \frac{2}{3} - \frac{1}{3} + \frac{1}{{21}} = \frac{{16}}{{21}}$
The volume of ${\cal W}$:
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} 1{\rm{d}}V = \mathop \smallint \limits_{x = - 1}^1 \mathop \smallint \limits_{y = {x^2}}^1 \mathop \smallint \limits_{z = 0}^{1 - {y^2}} 1{\rm{d}}z{\rm{d}}y{\rm{d}}x = \frac{{16}}{{21}}$
