Answer
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}V = \mathop \smallint \limits_{x = 0}^1 \mathop \smallint \limits_{y = 0}^{1 - x} \mathop \smallint \limits_{z = 1 - x - y}^{2 + {x^2} + {y^2}} f\left( {x,y,z} \right){\rm{d}}z{\rm{d}}y{\rm{d}}x$
Work Step by Step
We have a solid region ${\cal W}$ located below the paraboloid ${x^2} + {y^2} = z - 2$ and lies above the part of the plane $x+y+z=1$ in the first octant $x \ge 0$, $y \ge 0$, $z \ge 0$.
From the figure attached, we see that ${\cal W}$ can be described as a $z$-simple region bounded below by the plane $x+y+z=1$ and bounded above by the paraboloid ${x^2} + {y^2} = z - 2$. Thus,
$1 - x - y \le z \le 2 + {x^2} + {y^2}$
Referring to the figure attached, we see that the projection of ${\cal W}$ onto the $xy$-plane ($z=0$) is the domain ${\cal D}$ bounded by the line $x+y=1$ in the first quadrant. We choose to describe ${\cal D}$ as a vertically simple region defined by
${\cal D} = \left\{ {\left( {x,y} \right)|0 \le x \le 1,0 \le y \le 1 - x} \right\}$
Thus, the description of ${\cal W}$ is given by
${\cal W} = \left\{ {\left( {x,y,z} \right)|0 \le x \le 1,0 \le y \le 1 - x,1 - x - y \le z \le 2 + {x^2} + {y^2}} \right\}$
Now we write the triple integral $\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}V$ as an iterated integral:
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}V = \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal D}^{} \left( {\mathop \smallint \limits_{z = 1 - x - y}^{2 + {x^2} + {y^2}} f\left( {x,y,z} \right){\rm{d}}z} \right){\rm{d}}A$
$\mathop \smallint \limits_{}^{} \mathop \smallint \limits_{}^{} \mathop \smallint \limits_{\cal W}^{} f\left( {x,y,z} \right){\rm{d}}V = \mathop \smallint \limits_{x = 0}^1 \mathop \smallint \limits_{y = 0}^{1 - x} \mathop \smallint \limits_{z = 1 - x - y}^{2 + {x^2} + {y^2}} f\left( {x,y,z} \right){\rm{d}}z{\rm{d}}y{\rm{d}}x$
