0
$\begingroup$

I have the integral: ${\iiint} x^2 dx dy dz$ which is bounded from above by the elliptic paraboloid $z=2-x^2 - y^2$ and from below by the upper part of the cone $z^2 = x^2 + y^2$

I want to write this integral as a triple integral with cylindrical coordinates in the order $dz dr d\theta $

I know $r^2 = x^2 + y^2$, so by using this I can say $z^2 = x^2 + y^2$ becomes $z^2 = r^2$ (and so $z=r$) and $z=2-x^2 - y^2$ becomes $z=2-r^2$

I also set $r=2-r^2$ (since they both equal $z$) and found $r=1$ (since we want a positive $r$)

So I have what I believe is the correct set-up for the upper and lower bounds of my triple integration: ${\int}_0 ^{2\pi} {\int}_0 ^1 {\int}_r ^{2-r^2} *\text{something}* dz dr d\theta$

What I'm not sure of is what $*\text{something}*$ is, it was $x^2$ but I'm not sure how it changes now that my triple integral has cylindrical coordinates.

If someone could show me (or give me a hint to do it myself) what happens to $x^2$ when we convert to cylindrical coordinates it would really help.

I know what to do after I find out what $*\text{something}*$ is so there's no need to do the triple integral as well (unless you want to).

Thanks in advance

$\endgroup$
2
  • $\begingroup$ Why do you say elliptic paraboloid? It is not. All your working is based on $x = r \cos\theta, y = r\sin\theta$. That gives you value of $x^2$. $\endgroup$
    – Math Lover
    Jul 30 '21 at 14:13
  • $\begingroup$ In my first line I just typed the question I was given, in the question they say it's an elliptic paraboloid so I just assumed it to be true. To be honest I don't think it affects the question much so it's not too important. $\endgroup$
    – Charlie P
    Jul 30 '21 at 15:39
1
$\begingroup$

In cylindrical coordinates, $x^2$ becomes $r^2\cos^2\theta$. Besides, you have to multiply everything by the volume element, which is $r$. So, compute$$\int_0^{2\pi}\int_0^1\int_r^{2-r^2}r^3\cos^2(\theta)\,\mathrm dz\,\mathrm dr\,\mathrm d\theta.$$

$\endgroup$
4
  • $\begingroup$ Ok, I have 2 short questions. 1) When you say $x^2$ becomes $r^2 cos^2 (\theta)$, did you work this out or is it just known that this is the case? 2) Do we multiply everything by $r$ every time regardless of what we have in the middle? $\endgroup$
    – Charlie P
    Jul 30 '21 at 13:42
  • $\begingroup$ 1) It is universal. Working with cylindrical coordinates means that you are doing $x=r\cos(\theta)$ and $y=r\sin(\theta)$. 2) Yes. $\endgroup$ Jul 30 '21 at 16:26
  • $\begingroup$ Any more questions? $\endgroup$ Jul 31 '21 at 9:16
  • $\begingroup$ No that's perfect thanks. Sorry, I forgot to reply and upvote. Thanks for the help $\endgroup$
    – Charlie P
    Aug 2 '21 at 12:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.