Math Is Fun Forum

bossk171

Member

Registered: 2007-07-16

Posts: 305

Arc Length of a Parabola

Alright so arc length is:

Which is fairly strait forward assuming you know calculus. How ever if I want to find the arc length of the parabola y = x² I start to struggle.

The answer The Integrator gives is:

My answer to that is WHAT? MY knowledge of hyperbolic functions is pretty basic, but I'm familiar with them. Can any one explain where this comes from? At the very least, can you tell me how to evaluate it for a given a and b?

Thank you.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Arc Length of a Parabola

well... I assume that integrator just gives you the antiderivative, so you could just plug (a) into that expression, and subtract that from the result of plugging (b) into that expression.
In otherwords,

then the evaluation is (F(b) - F(a))

But somehow i think you'd know that!

I think they are using inverse hyperbolic trigonometric substitution to do that:

let sinh(u) = 2x

then:

4x^2 = sinh^2(u) and dx = (1/2) cosh(u) du

so we have:

Note I've replaced a and b with a* and b* to represent the values of u where x = a and x = b respectively. Now using the identity, sinh^2 + 1 = cosh^2, we get

now I don't really remember how you handle even powers of hyper-trig functions, but I'm pretty sure there's an identity somewhere that will simplify this.

Anyway, when you're done finding the antiderivative in terms of u, you resubstitute for u using the relation sinh(u) = 2x, or equivalently, sinh^-1(2x) = u, and then simplify. Once you replace the expression with one involving only x, then your limits of evaluation move back to a, and b (rather than a* and b*). I'm assuming what you were given by the calculator is what you get as a result.

Last edited by mikau (2008-05-31 14:21:45)

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A logarithm is just a misspelled algorithm.

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: Arc Length of a Parabola

Wait,

Last edited by Dragonshade (2008-05-31 14:30:50)

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Arc Length of a Parabola

Thanks for the help, but I guess I wasn't really clear... I don't know how to evaluate any of the hyperbolic functions (I don't know what sinh(2a) is) and that's more of the issue I'm having. My TI-83 is rejecting mixing trig functions with complex numbers. How do I evaluate it?

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: Arc Length of a Parabola

oooh, its quite simple. You can compute it easily in terms of e. (in fact, i think thats how they are defined)

the INVERSE of these functions, I'm not sure I remember. I'm pretty sure its an expression involving ln(x).

Last edited by mikau (2008-06-01 07:44:20)

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A logarithm is just a misspelled algorithm.

sunshine man

Guest

Re: Arc Length of a Parabola

lets make this a little more complex.

Do this for:

from d to e

soroban

Member

Registered: 2007-03-09

Posts: 452

Re: Arc Length of a Parabola

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