# Length of curve

• Dec 6th 2009, 03:26 PM
Todeezy
Length of curve
I'm getting stuck doing my integration for the following problem:

Find the length of the curve r(t) = ((2)^(1/2)t)i + (e^t)j + (e^-t)k

Sorry I'm not quite sure how to use the notation on this site yet but that first term is supposed to be a root 2 multiplied by t.
• Dec 6th 2009, 07:18 PM
HallsofIvy
Quote:

Originally Posted by Todeezy
I'm getting stuck doing my integration for the following problem:

Find the length of the curve r(t) = ((2)^(1/2)t)i + (e^t)j + (e^-t)k

Sorry I'm not quite sure how to use the notation on this site yet but that first term is supposed to be a root 2 multiplied by t.

The derivative of the vector function is $\displaystyle \sqrt{2}\vec{i}+ e^t\vec{j}- e^{-t}\vec{k}$ and its length is $\displaystyle \sqrt{2+ e^{2t}+ e^{-2t}}$. If you look closely at the terms inside the square root, you will see that they are $\displaystyle e^{2t}+ 2+ e^{-2t}= (e^{t})^2+ 2e^{t}e^{-t}+ (e^{-t})^2$$\displaystyle = (e^t+ e^{-t})^2$. That makes the integral very easy!