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Write out the triple integral for $f(x,y,z)$ to describe a region bounded by the planes $x=0, \, y=0,\, z=0, \, x+y=4$, and $x=z-y-1$.

I plugged in the 4th plane into the 5th plane and solved $z=5$. So, I know z is between 0 and 5 so it is the outermost integral.

I'm not entirely sure how to write out the bounds for x and y. Can someone help me with this?

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  • $\begingroup$ is the correct answer. x varies from 0 to 4. y varies from 0 to 4-x, and z varies from 0 to x+y+1? $\endgroup$
    – Math19384
    Commented Apr 8, 2018 at 0:36
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    $\begingroup$ Yes, @Math19384, that's right. $\endgroup$
    – Ted Shifrin
    Commented Apr 8, 2018 at 0:43
  • $\begingroup$ You ask too many questions (6 questions in 12 hours !) and questions that are correlated without mentionning the previous ones like here (math.stackexchange.com/q/2727004). Moreover, I have seen that you never validate/check questions with good answers. Slow down : one question a day is enough... and ask once you have really worked on it. $\endgroup$
    – Jean Marie
    Commented Apr 8, 2018 at 7:00
  • $\begingroup$ @JeanMarie i ask the questions i want to ask when i want to ask them. i try my best to attempt the problem before hand and show my progress or how far i've gotten for each question. $\endgroup$
    – Math19384
    Commented Apr 9, 2018 at 2:54
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    $\begingroup$ It's a rule of thumb. I wasn't aware of the "6 questions a day, etc." (in the case of the OP, it was even 6 questions within half a day)... but, I think that it's not a good habit to ask help for little difficulties that are resolved by simply working on them. My own experience is that I have been able to solve more and more difficult issues because I had to rely on myself without much help from outside... $\endgroup$
    – Jean Marie
    Commented Apr 11, 2018 at 20:08

1 Answer 1

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The lines $x = 0$, $y = 0$ and $x + y = 4$ form a right-angled triangle with vertices $(0,0), (4,0)$ and $(0,4)$ on the $xy$-plane. Therefore, $0 \le x \le 4$ and $0 \le y \le 4 - x$. Make $z$ the subject in the equation for the 5th plane: $z = x + y + 1$. Denote the region as $R$

$$\therefore \iiint_R f(x,y,z) \,dx dy dz = \int_0^4 \int_0^{4-x} \int_0^{x+y+1} f(x,y,z) \, dz dy dx$$

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  • $\begingroup$ The OP asks questions in a compulsive way, sometimes on the same subject like this one which is related, as you know, to this one asked some hours before :(math.stackexchange.com/q/2727004) $\endgroup$
    – Jean Marie
    Commented Apr 8, 2018 at 7:04
  • $\begingroup$ @JeanMarie OP knows the region of integration (say ❤-shaped), but he/she got stuck at expressing it at iterated integrals. $\endgroup$
    – GNUSupporter 8964民主女神 地下教會
    Commented Apr 8, 2018 at 17:51
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    $\begingroup$ [+1] You do not deserve a downvote at all ! $\endgroup$
    – Jean Marie
    Commented Apr 11, 2018 at 20:09
  • $\begingroup$ @JeanMarie Thank you ! $\endgroup$
    – GNUSupporter 8964民主女神 地下教會
    Commented Apr 11, 2018 at 20:14

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