Q:

a question about triple integral

In this question, is it right to let $x^2+z^2 = r^2$ and $x=r\cos \theta$, $z=r\sin \theta$?

A:

Let's introduce the following cylindrical coordinates
$$x=r\cos\theta,\ y=y, \ z=r\sin\theta.$$
Then, indeed $$r=\sqrt{x^2+z^2}.$$
The Jacobian of the substitution is $r$.
So after our integral transformation we have 
$$\iiint_Nr^2\ dydrd\theta.$$
The following figure helps identifying $N$:

Finally, we have
$$\int_0^{2\pi}\left[\int_0^2\ r^2\left[\int_{r^2}^41\ dy\right] dr\right]\ d\theta.$$

