Why Dropping the Sqrt{1 + 1/x^4} dx Matters for Surface Area

[woodysooner]

ok I know how to solve for the volume no prob, but when you find the surface area, in all the proof they always drop the root, because something about being bigger than one

Sqrt{1 + 1/x^4} dx

can someone explain why they drop that, and how it's possible to have finite volume and infinite surface area.

 

[TenaliRaman]

I particularly don't understand your first question.

as for ur second question, it seems you have done some good research noticing the title of this post :)

you may want to look here first,
http://mathforum.org/library/drmath/view/52017.html

in case anything abt is doubtful may be asked freely.

-- AI

 

[woodysooner]

That's the exact site i was on lol

Note that Sqrt{1 + 1/x^4} is always at least as big as 1, so we can drop it from the integral, and if the resulting integral diverges
to infinity.

this is on the site, and that what i was asking about, i remember doing these probs and we never ignored the root we always worked it out.

 

[Gza]

I haven't got a chance to read the actual proof, but what I do know is that when dealing with limits going to infinity, finite quantities have no relevance and can be excluded from any limit or integration problem.

 

[matt grime]

why is it surprising that you can infinite surface area and finite volume? knock it down a dimension and consider 1/x^2 between 1 and infinity and bound below by the x axis. perimeter is infinite and area is finite...

 

[woodysooner]

GZA, can you say that again somehow and explain that' s kewl but i didint know that.

 

[woodysooner]

btw matt, what would that look like.

 

[Corneo]

They droped $$\sqrt{1+x^4}$$ knowing that it is > 1. So Then it can be said that

$$\int_{1}^{\infty} 2 \pi \frac {1}{x} \sqrt {1+x^4} \geq \int_{1}^{\infty} 2 \pi \frac {1}{x}$$

The right hand side diverges because of p-series so by comparison, the integral on the left diverges also.

Just wondering I am trying to graph the picture of the horn on Maple 8, I think I determined the equation to be $$\frac {1}{x^2}=y^2+z^2$$. But Maple needs me to solve for z. So there is going to be some ugly square root. The horn looks pretty messed up from then. Anyone know what to do?

 

[mathwonk]

take a unit square which thus has unit area. Cut it in half and lay half of it down along the x axis. then cut the remaining half in half and lay the =resulting quarter down along the x axis. you now have a segment of length 2 units. Now go back to the remaining quarter of a square and cut it in half again, and lay the resulting half piece down along the x-axis next to the two previous pieces.

Do you see that you can continue this process forever, thus generating an infinite lonegth from a finite area? In the same way a finite volume can give rise to an infinite area.