Limits for triple integral parabolic cylinder
Solution 1
To check the set up we need to make some sketches of the domain as for example in $zx$ and $zy$ planes
and also important in the $xy$ plane for $z$ fixed that is a rectangular domain
from here your set up seems correct.
Solution 2
Yes, the limits are correct, the volume is given by the following iterated integral $$\int_{z=0}^{4}\left(\int_{x=\sqrt{z}}^{\sqrt{z}}\left(\int_{y=0}^{4z} dy\right) dx\right) dz.$$ Can you take it from here? Cartesian coordinates seems to be fine here.
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Pumpkinpeach
Updated on December 10, 2022Comments

Pumpkinpeach 11 months
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 $x = \sqrt{z},\dots, \sqrt{z}$,
$y= 0,\dots, 4z$, $z=0,\dots,4$.It seems too easy, should I be using polars?

Shaun almost 5 yearsPlease do not ask questions using pictures of text, since otherwise the question is difficult to search for and some users cannot see the pictures on some devices.

Robert Z almost 5 yearsFair question! (+1)

user almost 5 yearsIf it easy good for you! We don't really need polar in that case since for $z$ fixed the domain is rectangular.

Pumpkinpeach almost 5 yearsI mean its a lot of marks and I felt I hadn't done enough work for it.

Robert Z almost 5 years@Pumpkinpeach To avoid downvotes, let us know your progress. BTW please take a few minutes for a tour: math.stackexchange.com/tour

user almost 5 years@RobertZ That's a good advice but I don't think that the question posed was totally insufficient. The asker presented his work on that, even if in a short way. Anyway, I agree that the question can be improved adding some more detail abou the result obtained.

Robert Z almost 5 years@gimusi I agree, but unfortunately this question got 3 downvotes.

user almost 5 years@RobertZ Thise downvotes are not motivated in my opinion, even if I agree that the question can be improved, it doesn't deserve such kind of welcoming.
