Limits for triple integral parabolic cylinder

1,181

Solution 1

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.

Solution 2

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

1,181

Pumpkinpeach

Updated on December 10, 2022

Comments

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 $x = -\sqrt{z},\dots, \sqrt{z}$,
$y= 0,\dots, 4-z$, $z=0,\dots,4$.

It seems too easy, should I be using polars?
- Shaun almost 5 years
  
  Please 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 years
  
  Fair question! (+1)
- user almost 5 years
  
  If 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 years
  
  I 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.