I need to calculate the integral:
$$\iiint_E \sqrt{x^2+z^2}\,dV$$
such that $E$ is the solid bounded by $y=x^2+z^2$ and $y=4$.
I am not able to decide the limits of integration, a walk through for this would be really appreciated. Additionally, is there a way I can solve this using cylindrical co-ordinates?
HINT
Let use cylindrical coordinates with
$x=r\cos \theta$
$z=r\sin \theta$
$y=y$
$dV=r \,dy\, dr\, d\theta$
then
$$\iiint_E \sqrt{(x^2+z^2)}.dV=\int_{0}^{2\pi}d\theta\int_0^4 dy\int_0^{\sqrt y}r^2 \, dr$$
In cartesian coordinates we have
$$\iiint_E \sqrt{(x^2+z^2)}.dV=\int_{0}^{4}dy\int_{-\sqrt y}^{\sqrt y} dx\int_{-\sqrt{y-x^2} }^{\sqrt{y-x^2}}\sqrt{x^2+z^2}\, dz$$