I have the function $\displaystyle f(x) = x^2$, is there a way to measure the length of the line between $\displaystyle (0,0)$ and $\displaystyle (2,4)$ ?

Printable View

- Jun 2nd 2006, 01:49 AMchanceyLength of a curve?
I have the function $\displaystyle f(x) = x^2$, is there a way to measure the length of the line between $\displaystyle (0,0)$ and $\displaystyle (2,4)$ ?

- Jun 2nd 2006, 02:44 AMQuickSpecify
Are you trying to find the distance between the two points or the length of the curved line along the two points?

- Jun 2nd 2006, 03:12 AMchanceyQuote:

Originally Posted by**Quick**

- Jun 2nd 2006, 03:15 AMTD!
Yes, that's possible with integration.

The arc length of a function f(x) between x = a and x = b is given by:

$\displaystyle \ell = \int\limits_a^b {\sqrt {1 + f'\left( x \right)^2 } dx} $

In your case (I omitted the calculation), the result is:

$\displaystyle

\ell = \int\limits_0^2 {\sqrt {1 + 4x^2 } dx} = \frac{1}{4}\ln \left( {\sqrt {17} + 4} \right) + \sqrt {17} \approx 4.647

$ - Jun 2nd 2006, 03:22 AMchanceyQuote:

Originally Posted by**TD!**

Thanks - Jun 2nd 2006, 03:24 AMTD!
- Jun 2nd 2006, 03:30 AMchancey
OK thanks, I'll give it a read

- Jun 2nd 2006, 03:51 AMCaptainBlackQuote:

Originally Posted by**chancey**

RonL