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John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Curvy line length calculus?

You know how calculus can find area under a curve?
Well I was wondering if we could extend this idea
to invent a calculus to finding the length of
the curvy or straight line of a function.
The tangent function isn't exactly right, but it sort of
gives the idea because a function with a large
slope makes a long line between two values of x,
however, if the line is horizontal (like y = 5), then
the length is just difference of two x values, but
the slope is zero, so tangent isn't working here.
(Off the subject, what is tanh (hyperbolic) all about?)

What about pythagoreans theorm, would that help to
find the incremental length at each point based on
√ (dy² + dx²) ???

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igloo myrtilles fourmis

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Curvy line length calculus?

Say you want the length of the line on a parabola

from
x = 0 to x=5.
Perhaps

, I am not good at calculus yet.
Maybe Length of line =

Last edited by John E. Franklin (2005-12-11 06:00:08)

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igloo myrtilles fourmis

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Curvy line length calculus?

This is probably all wrong, but I am learning to use LaTeX!!!

Even if this is wrong, I don't know how to integrate it? 
Anyone know where I can learn this one?
...
I found "Power Substitution" on internet.
I might learn this!
...
Well If I let

, it seems like a good
start, but then

gets just as messy as what I
started with, so that's no help...

Last edited by John E. Franklin (2005-12-11 05:59:33)

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igloo myrtilles fourmis

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Curvy line length calculus?

I cheated and used a standard mathematical tables integrals answer.
Plugging in 5 and 0, and subtracting revealed a line length of 26.40253635
This might actually be right, since it is a little longer than going
straight from (0,0) to (5,25), and that's because it is a parabola.
Maybe it's right!  Hurray!  Maybe it's not.  I'm not sure yet.

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igloo myrtilles fourmis

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Curvy line length calculus?

What is the length of the curvy line of a sine wave graphed from 0 to 2π radians?
I'll work on this and wish me luck...:)

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igloo myrtilles fourmis

Flowers4Carlos

Member

Registered: 2005-08-25

Posts: 106

Re: Curvy line length calculus?

you are absolutely correct john!  the integral for finding the distance on a curve is how you explained it!  although it's a common formula, i find it smart of you (and sooo cute!!!!) that you were able to figure it out on your own!!  so smart!!!!
http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/moreApps/arclength.html

as for ∫(1+4x²)^(1/2)dx you either gotta use trigonometric substitution or a table chart (which is also fine).  if you want i can solve it "manually" for you... it's not gonna be pretty though.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Curvy line length calculus?

Thank you very much for the compliment.
You made my day!
I'll check out that link in a minute.
And yes, I did find it in a table, and I got 26.40253635
for the length from 0 to 5 for

I am still struggling with how calculus works, and it is
of great interest to me because I sometimes have to
resort to doing computer approximations by breaking
the problem into the tiny pieces and adding it up.
Newton was a

for sure!

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igloo myrtilles fourmis