# How to reparametrize by arc length?

• October 28th 2012, 02:40 PM
kandygirl16
How to reparametrize by arc length?
1. Find the length of the helix f(t)= [(cost)^2), (sint)^2, 2t^2] for t E [0,10]. Reparametize this curve by arc length.
• October 28th 2012, 03:28 PM
TheEmptySet
Re: How to reparametrize by arc length?
Quote:

Originally Posted by kandygirl16
1. Find the length of the helix f(t)= [(cost)^2), (sint)^2, 2t^2] for t E [0,10]. Reparametize this curve by arc length.

To find the length you need to integrate

$s=\int_{a}^{b}\sqrt{\left( \frac{dx}{dt}\right)^2+\left( \frac{dy}{dt}\right)^2+\left( \frac{dz}{dt}\right)^2}}}, \quad t \in [a,b]$

This gives

$s=\int_{0}^{10}\sqrt{4\sin^2(t)\cos^2(t)+4\sin^2(t )\cos^2(t)+16t^2}dt}$

This function does not have a "nice" anti derivative.

Did you mean the helix $\mathbf{r}(t)=<\cos(t),\sin(t),2t>$?
• October 28th 2012, 03:58 PM
kandygirl16
Re: How to reparametrize by arc length?
Hi, thanks for the reply. No unfortunately, this is the one I meant.