Answer
(a) meaningful triple integral
(b) not a meaningful triple integral
Work Step by Step
(a) We have $\mathop \smallint \limits_0^1 \mathop \smallint \limits_0^x \mathop \smallint \limits_{x + y}^{2x + y} {{\rm{e}}^{x + y + z}}{\rm{d}}z{\rm{d}}y{\rm{d}}x$.
1. Inner integral: $\mathop \smallint \limits_{x + y}^{2x + y} {{\rm{e}}^{x + y + z}}{\rm{d}}z$
The inner integral is a function of $x$ and $y$:
$g\left( {x,y} \right) = \mathop \smallint \limits_{x + y}^{2x + y} {{\rm{e}}^{x + y + z}}{\rm{d}}z$
2. Middle integral: $\mathop \smallint \limits_0^x g\left( {x,y} \right){\rm{d}}y$
The middle integral is a function of $x$ alone:
$h\left( x \right) = \mathop \smallint \limits_0^x g\left( {x,y} \right){\rm{d}}y$
3. The outer integral: $\mathop \smallint \limits_0^1 h\left( x \right){\rm{d}}x$
The outer integral is evaluated with respect to $x$, so it is a meaningful triple integral.
(b) We have $\mathop \smallint \limits_0^1 \mathop \smallint \limits_0^z \mathop \smallint \limits_{x + y}^{2x + y} {{\rm{e}}^{x + y + z}}{\rm{d}}z{\rm{d}}y{\rm{d}}x$.
1. Inner integral: $\mathop \smallint \limits_{x + y}^{2x + y} {{\rm{e}}^{x + y + z}}{\rm{d}}z$
The inner integral is a function of $x$ and $y$:
$g\left( {x,y} \right) = \mathop \smallint \limits_{x + y}^{2x + y} {{\rm{e}}^{x + y + z}}{\rm{d}}z$
2. Middle integral: $\mathop \smallint \limits_0^z g\left( {x,y} \right){\rm{d}}y$
The integral is evaluated with respect to $y$, holding $x$ constant. However, because there is a $z$ variable at the upper limit, it is also a function of $z$:
$h\left( {x,z} \right) = \mathop \smallint \limits_0^z g\left( {x,y} \right){\rm{d}}y$
The integral does not make sense. Without further evaluating the outer integral, we conclude that it is not a meaningful triple integral.
