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

In summary, the conversation discusses the dropping of the root in the integral Sqrt{1 + 1/x^4} dx, which is justified by the fact that the function is always greater than 1. This leads to a divergence in the integral, which is explained using a comparison to the p-series. The conversation also briefly mentions the graphing of a horn and how a finite volume can result in an infinite surface area.
  • #1
woodysooner
174
0
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.
 
Physics news on Phys.org
  • #2
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
 
  • #3
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.
 
  • #4
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.
 
  • #5
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...
 
  • #6
GZA, can you say that again somehow and explain that' s kewl but i didint know that.
 
  • #7
btw matt, what would that look like.
 
  • #8
They droped [tex]\sqrt{1+x^4}[/tex] knowing that it is > 1. So Then it can be said that

[tex]\int_{1}^{\infty} 2 \pi \frac {1}{x} \sqrt {1+x^4} \geq \int_{1}^{\infty} 2 \pi \frac {1}{x} [/tex]

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 [tex]\frac {1}{x^2}=y^2+z^2[/tex]. 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?
 
  • #9
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.
 

1. Why is it important to drop the sqrt{1 + 1/x^4} dx when calculating surface area?

Dropping the sqrt{1 + 1/x^4} dx is important because it allows for a more accurate calculation of the surface area. This term is known as the "Jacobian" and it takes into account the change in variables when going from Cartesian to polar coordinates. If this term is not dropped, it can lead to an overestimation of the surface area.

2. How does dropping the sqrt{1 + 1/x^4} dx affect the overall surface area calculation?

Dropping the sqrt{1 + 1/x^4} dx can significantly impact the overall surface area calculation. It can result in a smaller surface area, as the Jacobian term accounts for the change in variables and can lead to an overestimation if not dropped. This can be especially important in precise measurements and calculations.

3. Can dropping the sqrt{1 + 1/x^4} dx affect the accuracy of the surface area calculation?

Yes, dropping the sqrt{1 + 1/x^4} dx can affect the accuracy of the surface area calculation. This term is important in accurately representing the change in variables and without it, the calculation may not be as precise. Therefore, it is important to properly drop this term to ensure accuracy in the surface area calculation.

4. What other situations may require dropping the sqrt{1 + 1/x^4} dx in surface area calculations?

Dropping the sqrt{1 + 1/x^4} dx may also be necessary in other situations where a change in variables occurs, such as when calculating surface area in spherical or cylindrical coordinates. In these cases, the Jacobian term would need to be dropped to accurately calculate the surface area.

5. How can not dropping the sqrt{1 + 1/x^4} dx affect the overall results of a surface area experiment?

Not dropping the sqrt{1 + 1/x^4} dx can lead to significant errors in the results of a surface area experiment. It can cause an overestimation of the surface area, which can impact the overall conclusions and findings of the experiment. Therefore, it is crucial to properly drop this term in order to obtain accurate results in surface area experiments.

Similar threads

  • Calculus
Replies
29
Views
514
Replies
4
Views
188
  • Calculus
Replies
1
Views
1K
Replies
2
Views
2K
Replies
2
Views
991
  • Classical Physics
Replies
33
Views
2K
  • Calculus and Beyond Homework Help
Replies
11
Views
2K
Replies
2
Views
1K
Replies
8
Views
2K
Back
Top