Length of a curve - A calculus approach

In summary, the conversation discusses a 16-year-old high school student's love for math and their creation of a formula for calculating the length of a graph curve. They seek help in evaluating the limit for the formula, and receive guidance on how to factor out the dx term and use the textbook formula for arc length. They also receive recommendations for online resources for learning calculus.
  • #1
Levis2
43
0
Now I'm a 16-year old high school student, and as some of you might know, i like math :) I have been studying the integration by riemann sums lately, and i truly love the logical concept. So I decided to create my own formula for calculation of graph curve length (without looking at the present solutions - this i consider cheating, and i only do this if i find the task impossible .. :) and after 30 min i ended up with a formula, that works (i think). I can use it to calculate curve lengtt manually by doing it in hand, and choose a certain number of rectangles n=some number. But after i confirmed that it works, i wanted to evaluate the limit for n-->infinity, so the true length shone through. This was my original thought with the formula .. I ended up hitting a dead end though. I am simply not able to evaluate the limit...

I don't know if it's because my skills are good enough, or if it's because it's impossible to do so :) So i need a bit of help. This is how the formula looks like;
(it's enclosed in the word document) - i would appreciate if anyone would write in a post, or tell me how to write in the post .. :) It's annoying for you to download the formula

I have tried evaluating for a both a linear function and a non-linear .. I can't seem to find the limit. Maybe it doesn't exist? Maybe my formula is completely wrong? I don't know :P

thx
 

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  • arc length.doc
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  • #2
Your formula looks good. (Nice work on your own).

As far as finding the limit, its going to be rather messy. Remember in the Riemann integrals the differential [dx] term corresponds to the interval lengths [Delta x] corresponding to your (x2-x1)/n factors.

You can factor their squares out of the sum inside the radical and then taking the square root you'd get something like:

[tex] \sum_i \sqrt{1+ [f'(x_i)]^2} \Delta x[/tex]
which becomes in the limit:
[tex] \int_{x_1}^{x_2} \sqrt{1+[f'(x)]^2} dx[/tex]
That's the textbook formula you find for arc length of a function's curve.

You may find this makes more sense, and generalizes nicely to parametric curves if you work in terms of differentials.
 
  • #3
The expression you have when taken to the limit can be expressed as:
∫(dx2 + dy2)1/2 = ∫(1 + (dy/dx)2)1/2dx
where y=f(x). This is the usual expression for curve length.
 
  • #4
jambaugh said:
Your formula looks good. (Nice work on your own).

As far as finding the limit, its going to be rather messy. Remember in the Riemann integrals the differential [dx] term corresponds to the interval lengths [Delta x] corresponding to your (x2-x1)/n factors.

You can factor their squares out of the sum inside the radical and then taking the square root you'd get something like:

[tex] \sum_i \sqrt{1+ [f'(x_i)]^2} \Delta x[/tex]
which becomes in the limit:
[tex] \int_{x_1}^{x_2} \sqrt{1+[f'(x)]^2} dx[/tex]
That's the textbook formula you find for arc length of a function's curve.

You may find this makes more sense, and generalizes nicely to parametric curves if you work in terms of differentials.

well, i feel very, very stupid right now - how am i suppose to factor out the the dx term ? :P I have only received 3 months worth of lessons in very basic algebra, since I am fresh out of 9nth grade :) I haven't even had a single lesson about functions lol. I don't see how i can factor out the dx term?

And yea i know my ((x2-x1)/n) term is equivalent to dx in the summs, but i prefer to write it the other way to give it a personal touch i guess :)

I apologise for any possible errors in notation, but since all of this is self-study, it takes a while to get familiar with the correct notations :)

But if that's the original formula for arch length, then i have actually re-invented the formula ?:P
 
  • #5
hey if your looking to get ahead of your class a bit a great site for you is www.khanacademy.org ... goes from secondary school maths right up to 1 year of college maths.
 
  • #6
I teach Calculus
Yes, Khan Academy is a great site for learning online by video.
Also Google for "calculus university houston" and find 53 excellent Calc videos

But be careful. You need to be able to learn by reading from a text too.
Videos do not exist [yet] for all topics or all subjects.

I wish my 9th graders had half the initiative and potential you clearly have.

Good luck
 
  • #7
paulfr said:
I teach Calculus
Yes, Khan Academy is a great site for learning online by video.
Also Google for "calculus university houston" and find 53 excellent Calc videos

But be careful. You need to be able to learn by reading from a text too.
Videos do not exist [yet] for all topics or all subjects.

I wish my 9th graders had half the initiative and potential you clearly have.

Good luck
You teach Calc to 9th graders?
 
  • #8
Levis2 said:
well, i feel very, very stupid right now - how am i suppose to factor out the the dx term ? :P I have only received 3 months worth of lessons in very basic algebra, since I am fresh out of 9nth grade :) I haven't even had a single lesson about functions lol. I don't see how i can factor out the dx term?

And yea i know my ((x2-x1)/n) term is equivalent to dx in the summs, but i prefer to write it the other way to give it a personal touch i guess :)

I apologise for any possible errors in notation, but since all of this is self-study, it takes a while to get familiar with the correct notations :)

But if that's the original formula for arch length, then i have actually re-invented the formula ?:P

There are a few details I'd like to mention. The algebra of it is:
[tex] \sqrt{X^2A + X^2B} = \sqrt{X^2(A+B)} = \sqrt{X^2} \cdot \sqrt{A+B}[/tex]
one uses the algebra fact the square root of a product is the product of the square roots, (and of course the distributive law to factor the sum).

Here's the tiny detail with which one should be careful:
[tex]\sqrt{X^2} = |X| [/tex]
not simply X.

What this means is you should get the right arc-length as long as you integrate in the positive direction i.e. x2 > x1. But if you reverse the integration (which changes the sign) the dx (incorrectly) becomes negative giving you a negative arclength. We should be more careful to put in the absolute value |dx| in the integral. But |dx| = dx if dx>0 and |dx| = -dx if dx<0. So we break it up into cases. If we integrate from larger x value to smaller (which gives us a negative value) we change the sign (to get back the positive arclength).

Or we can simply write:
[tex]S =\left| \int_{x_1}^{x_2}\sqrt{1+f'^2(x)}dx \right| [/tex]

Most texts don't bother but there's the chance a student will occasionally integrate in a negative direction (say by using a substitution) and wonder why they keep getting a sign error. It's really an error in the formula in neglecting this |dx| business.

Again, good work on that derivation. Keep it up and you'll go far. Oh, and there were no errors in notation, only that you choice not to use it.
 
  • #9
TylerH said:
You teach Calc to 9th graders?

No but I can see how you concluded that.
I teach 12th graders Calculus.
My 9th graders are about the same age as the OP author.
I wish more of them were as curious as this fellow is.
Unfortunately it is an expensive private school and many
come from wealthy homes and are not motivated much less curious.
Sigh ...
 
  • #10
jambaugh said:
There are a few details I'd like to mention. The algebra of it is:
[tex] \sqrt{X^2A + X^2B} = \sqrt{X^2(A+B)} = \sqrt{X^2} \cdot \sqrt{A+B}[/tex]
one uses the algebra fact the square root of a product is the product of the square roots, (and of course the distributive law to factor the sum).

Here's the tiny detail with which one should be careful:
[tex]\sqrt{X^2} = |X| [/tex]
not simply X.

What this means is you should get the right arc-length as long as you integrate in the positive direction i.e. x2 > x1. But if you reverse the integration (which changes the sign) the dx (incorrectly) becomes negative giving you a negative arclength. We should be more careful to put in the absolute value |dx| in the integral. But |dx| = dx if dx>0 and |dx| = -dx if dx<0. So we break it up into cases. If we integrate from larger x value to smaller (which gives us a negative value) we change the sign (to get back the positive arclength).

Or we can simply write:
[tex]S =\left| \int_{x_1}^{x_2}\sqrt{1+f'^2(x)}dx \right| [/tex]

Most texts don't bother but there's the chance a student will occasionally integrate in a negative direction (say by using a substitution) and wonder why they keep getting a sign error. It's really an error in the formula in neglecting this |dx| business.

Again, good work on that derivation. Keep it up and you'll go far. Oh, and there were no errors in notation, only that you choice not to use it.

Haha that's rather obvius .. weird i didn't see the fact that i could just factor out dx :)

Now i tried to evaluate the limit of f(x)=x^2, from 0-5. I ended up this far;

(in the document .. sorry :)

And when i solved it on maple, it seemed correct. Hmm maybe I'm missing something elementary ? :) Its hard to study up on everything when one has to do it on their own.

Any suggestions ?
 

Attachments

  • ny limit.doc
    46.5 KB · Views: 213
  • #11
paulfr said:
I teach Calculus
Yes, Khan Academy is a great site for learning online by video.
Also Google for "calculus university houston" and find 53 excellent Calc videos

But be careful. You need to be able to learn by reading from a text too.
Videos do not exist [yet] for all topics or all subjects.

I wish my 9th graders had half the initiative and potential you clearly have.

Good luck

yeah i do prefer books as well - i think watching a lesson on video is very boring :) I would either be there in person or just read a book. It's a good skill to master too, since the majority of all the wisdom is this world is written in books hehe.

and thanks for the comment - although my intelligence level is rather average actually :)
 

1. What is the definition of "length of a curve" in calculus?

In calculus, the length of a curve refers to the distance along a curved line between two points. It is calculated using integration and is an important concept in determining the overall shape and characteristics of a curve.

2. How is the length of a curve calculated using calculus?

The length of a curve is calculated using a process called integration. This involves breaking the curve into infinitesimally small segments and adding them up using a mathematical formula. The result is an approximation of the total length of the curve.

3. What is the significance of calculating the length of a curve in calculus?

Calculating the length of a curve is important in understanding the shape and characteristics of a curve. It is also useful in various applications, such as finding the distance traveled by an object following a curved path or determining the amount of material needed to create a specific curved shape.

4. Are there any limitations to using calculus to calculate the length of a curve?

While calculus is a powerful tool for determining the length of a curve, there are limitations to its accuracy. The process of integration involves approximations, so the calculated length may not be exact. Additionally, complex curves may require advanced techniques and can be challenging to calculate using calculus.

5. How is the concept of the length of a curve used in real-world applications?

The concept of the length of a curve is used in a variety of real-world applications, including engineering, physics, and mathematics. It can be used to model and analyze various natural and man-made phenomena, such as the shape of waves, the trajectory of a projectile, or the design of roller coasters.

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