# Triple integral problem: cylindrical coordinates

1. Jul 25, 2013

### jackscholar

1. The problem statement, all variables and given/known data
I have a graph 1/x^2=y^2+z^2 where z=rsin(θ) and y=rcos(θ)
where 0≤r≤1 and 0≤θ≤2∏ on the zy-plane

The end result is attached (sorry, I'm not aware of how to use Latex :[ )
I can kind of understand how they determined the first bounds for the integral: the lowest x co-ordinate is 0 and the maximum co-ordinate is always 1/r. I can also kind of understand how they determined the second integral bounds, the maximum possible value for r is 1 and the lowest possible value for r is 0. For the angle the largest possible angle is 2∏ whereas the lowest possible angle is 0. I do not, however, understand why there is a half in front of the triple integral.

#### Attached Files:

• ###### Triple integral final equation.png
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2. Jul 25, 2013

### tiny-tim

hi jackscholar!

i don't understand what the question is …

is it to find a volume, if so of what?
surely it's from 1 to 1/r ?

3. Jul 25, 2013

### jackscholar

I am trying to determine the volume for Torricelli's Trumpet. I've done it the conventional way but have been want to prove it using the triple integral method. I stumbled across a website which stated a few things and got to the answer but it didn't explain how. I have proved the surface area using a different method to the disk method and would like to prove the volume using a different method but I'm not too good at triple integrals. The website is below, what I was looking at was on page 18 near the bottom and 19 near the top. Could you help m interpret what they are trying to do?

This is the website: http://www.palmbeachstate.edu/honors/documents/jeansergejoseph.pdf

4. Jul 25, 2013

### tiny-tim

5. Jul 25, 2013

### dirk_mec1

It's easy: it's just the volume of ONE of the horns.

6. Jul 25, 2013

### jackscholar

How would I calculate the volume of one of the horns using a triple integral, though?

7. Jul 25, 2013

### SteamKing

Staff Emeritus
8. Jul 25, 2013

### vanhees71

You use the cylindrical coordinates as given in #1:
$$\vec{r}=(x,r \cos \varphi,r \sin \varphi).$$
Then the boundary surface for one half of the horn is given as
$$x=\frac{1}{\sqrt{y^2+z^2}}=\frac{1}{r}.$$
Now the volume element in cylinder coordinates is
$$\mathrm{d}^3 \vec{r}=\mathrm{d} r \mathrm{d} \varphi \mathrm{d} x r.$$

Now using the definition for Gabriel's horn

http://en.wikipedia.org/wiki/Gabriel's_Horn

It's volume is given by
$$V=\int_0^1 \mathrm{d} r \int_0^{\varphi} \mathrm{d} \varphi \int_{1}^{1/r} \mathrm{d} x r.$$
Now you can easily evaluate the integral yourself.