Limits for triple integral parabolic cylinder

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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

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

enter image description here

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.

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Pumpkinpeach
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Updated on December 10, 2022

Comments

  • Pumpkinpeach
    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
      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
      Robert Z almost 5 years
      Fair question! (+1)
    • user
      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
      Pumpkinpeach almost 5 years
      I mean its a lot of marks and I felt I hadn't done enough work for it.
    • Robert Z
      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
      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
      Robert Z almost 5 years
      @gimusi I agree, but unfortunately this question got 3 downvotes.
    • user
      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.