Are the new limits of this triple integral fine?
$\iiint_E\, (x^2 + y^2 + z^2)\, dx\, dy\, dz$ where $E: \{(x,y,z) | x,y,z\geq0, x+y+z \leq a\}$, $a$ is positive constant
Under the transformation $$x+y+z=u$$
$$x+y=uv$$
$$x=uvw$$
$$\Rightarrow x=uvw, \, y= uv(1-w), \, z=u(1-v)$$
$$ \{(x,y,z) | x,y,z\geq0, x+y+z \leq a\} \Leftrightarrow \{(u,v,w) | 0\leq u\leq a, 0\leq v \leq 1, 0\leq w \leq 1\}$$
$$\iiint_E\, (x^2 + y^2 + z^2)\, dx\, dy\, dz \Leftrightarrow \int_{w=0}^{1}\,\int_{v=0}^{v=1}\int_{u=0}^{a} \, F(u,v,w)\,du\,dv\,dw$$
Is this correct?