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please excuse me if I'll say some dumb things but I'm not very informed in this topic.
If I have an hyperboloid $x^2+y^2-0.12z^2=9$ and I want to find the volume between $z=0$ and $z=15$, I can integrate it like this $ \int_{0}^{15}{AreaOfCircle(z)dz} $ where Area of Circle is $\pi(9+0.12z^2)$.
My problem is that I haven't really understood how to use triple integrals with cylindrical coordinates.
I know that the formula is $ \iiint{r*f(cylindrical coordinates)}drdzd\theta $, but in this case I've read that even if $f(cylindrical coordinates)=r^2-0.12z^2-9$ only the integration limits change, and in the triple integral I have to put only $r$ --> $\int_0^{2\pi}\int_0^{15}\int_0^{\sqrt{9+0.12z^2}}rdrdzd\theta$ instead of $\int_0^{2\pi}\int_0^{15}\int_0^{\sqrt{9+0.12z^2}}(r^2-0.12z^2-9)rdrdzd\theta$.
I don't understand why. Thanks.

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The volume of a 3-dimensional region is described by $$\iiint_{E}\,dV.$$ The statement about switching to cylindrical coordinates says that: If $T$ is a 3-dimensional region, and $T'$ is the same region described in cylindrical coordinates then $$\iiint_{T}f(x,y,z)\,dV = \iiint_{T'} rf(z,r,\theta)\,dzdrd\theta$$ Therefore, since you have a volume: $f(x,y,z)=1$ and $$\iiint_{E}\,dV = \iiint_{E'}r\,dzdrd\theta$$ where $E'$ is $E$ but in cylindrical coordinates

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  • $\begingroup$ Thanks. Sorry but I don't understand why my volume is f(x,y,z)=1 $\endgroup$ Jul 18 '20 at 17:39
  • $\begingroup$ Your volume is the triple integral, the function under the triple integral describes something more , usually a "density" in the case of a triple integral. Take some simpler examples: let $a>0$: then $\int_{0}^{a}\,dx=a$ (the length), $\int_{0}^{a}$ $\int_{0}^{a}\,dxdy=a^{2}$ (the area) and $\int_{0}^{a}\int_{0}^{a}\int_{0}^{a}\,dxdydz = a^{3}$ (the volume $\endgroup$ Jul 18 '20 at 17:49
  • $\begingroup$ Ok, so If I have to calculate a volume I have no function under the triple integral, right? $\endgroup$ Jul 18 '20 at 17:59
  • $\begingroup$ Yes, or rather you have a $1$ under the triple integral. $\endgroup$ Jul 18 '20 at 18:02
  • $\begingroup$ Ok, thanks! Now I've understood. $\endgroup$ Jul 18 '20 at 18:04

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