# Arc length of a curve represented by vector function

• Nov 28th 2009, 09:14 AM
mkelly09
Arc length of a curve represented by vector function
I'm having some trouble doing this question where i'm supposed to find the arc length of a curve represented by a vector function.

Question is: Find the arc length of the curve.

$\displaystyle r(t) = (\sqrt(2)*t) i + (e^t)j + 1/(e^t)k$ over the interval $\displaystyle 0<=t<=1$

So since arc length = int(0,1)[|r(t)|] i calculate r(t):
$\displaystyle r(t) = \sqrt(2)i + (e^t)j + -1/(e^t)k$
and
$\displaystyle |r(t)| = \sqrt(2 + (e^t)^2 + -1/(e^t)^2 )$
and this is where i get stuck. I must have to plug something into |r(t)| equation above for t to calculate |r(t)| and then plug it into the arc length equation. How am i supposed to integrate |r`(t)| as it is shown in the above form?
• Nov 28th 2009, 11:13 AM
Scott H
Quick correction:

$\displaystyle |\mathbf{r'}(t)|=\sqrt{2+(e^t)^2+\frac{1}{(e^t)^2} }.$

The integral

$\displaystyle \int_0^1\sqrt{2+e^{2t}+e^{-2t}}\,dt$

may be found by substitution. Hint: let $\displaystyle t=\frac{1}{2}\ln u$.