Limits for triple integral parabolic cylinder

Question

Determine the volume bounded by the parabolic cylinder $z=x^2$ and the planes $y=0$ and $y+z=4$.

My work. I am not sure if I have the correct limits for this question. I used $x = -\sqrt{z},\dots, \sqrt{z}$,
$y= 0,\dots, 4-z$, $z=0,\dots,4$.

It seems too easy, should I be using polars?

Answer 1

Yes, the limits are correct, the volume is given by the following iterated integral $$\int_{z=0}^{4}\left(\int_{x=-\sqrt{z}}^{\sqrt{z}}\left(\int_{y=0}^{4-z} dy\right) dx\right) dz.$$ Can you take it from here? Cartesian coordinates seems to be fine here.

Answer 2

To check the set up we need to make some sketches of the domain as for example in $z-x$ and $z-y$ planes

and also important in the $x-y$ plane for $z$ fixed that is a rectangular domain

from here your set up seems correct.