Triple integral problem: cylindrical coordinates

[jackscholar]

Homework Statement

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.

 

[tiny-tim]

hi jackscholar!

i don't understand what the question is …

is it to find a volume, if so of what?

the lowest x co-ordinate is 0 and the maximum co-ordinate is always 1/r

surely it's from 1 to 1/r ?

 

[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

 

[tiny-tim]

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

sorry, I've no idea how he got that integral

 

[dirk_mec1]

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

 

[jackscholar]

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

 

[SteamKing]

If you were using polar coordinates and trying to find the area of a figure, what form would the integral expression take?

Hint: http://tutorial.math.lamar.edu/Classes/CalcII/PolarArea.aspx

 

[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.