Please check my substitution for this integral, $$\iiint _V\left(x^2+z^2\right)\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z,$$ where $z=x^2+y^2$, $z\in\left[2,4\right]$.
I am not sure how to do the correct substitution, this is how I've done it since $z=x^2+y^2$ its a paraboloid:
$$x=\rho \cos\theta,\quad y=\rho \sin\theta,$$
where $\rho \in \left[\sqrt{2},2\right]$, $\theta \in \left[0,2\pi \right]$, $z\in \left[2,4\right]$.
No need to evaluate the integral, I just wanna check if I did the boundary right.
I am assuming you are using cylindrical coordinates. No the bounds are not correct. These bounds will give you volume between two cylinders of radii $\sqrt2 \ $ and $ \ 2$ of height $2$.
Also bounds depend on the order of the integral. So always specify that.
In cylindrical coordinates, if we are going in the order $dr$ first and then $dz$ and $d\theta$,
We have $z = x^2 + y^2 = r^2 \implies r = \sqrt z$ on paraboloid surface. So
$0 \leq r \leq \sqrt z$
$2 \leq z \leq 4$ and $0 \leq \theta \leq 2\pi$.