Calculating the Length of a Function

[daniel_i_l]

My friend told me that they had just learned an equation to find the length of a function. I decided that it would be cool to try to find it myself. I got: $$L(x) = \int \sqrt(f'(x)^2 +1)dx$$

I got that by saying that the length of a line with a slope of a over a distance of h is: $$\sqrt(f'(x)^2 +1)$$
Am I right?

 

[TD]

In general, when a function f is determined by a vectorfunction (so you have a parameter equation of the curve), the arc length is given by:

$$\ell = \int_a^b {\left\| {\frac{{d\vec f}}<br /> {{dt}}} \right\|dt}$$

There are of course conditions such as df/dt has to exist, be continous, the arc has to be continous.
Now when a function is given in the form "y = f(x)" you can choose x as parameter and the formula simplifies to:

$$\ell = \int_a^b {\sqrt {1 + y'^2 } dx}$$

Which is probably what you meant

 

[daniel_i_l]

Thanks!

 

[James R]

You're talking about arc length, right?

 

[TD]

Yes, at least that's what I assumed.