It seems like I am not using the good process to compute the length of a given parametric curve. I am not sure if it's inside my calculations or if the steps I use are not correct.
The equation of the curve is : $$ \vec{r}(t) = \sqrt{2}t\space\vec{i}+e^t\space\vec{j}+ e^{-t}\space\vec{k}, \space t = [0,1] $$
I am first starting by finding the derivative. $$ \vec{r}\space'(t) = \sqrt{2} \space \vec{i} + e^t \vec{j} - e^{-t} \vec{k}, \space t = [0,1] $$
After, I am calculating the magnitude of the last vector.
$$ ||\vec{r}\space'(t)|| = \sqrt{2(1+e^{t^2})},\space t = [0,1] $$
Finally, with
$$ L=\int_0^1 \sqrt{2(1+e^{t^2})}\;dt\\ $$
I find,
$$ L = \frac{\sqrt{2}}{2\sqrt{1+e}}-\frac{1}{2} $$
Which is a negative result and then impossible. Are my calculations wrong or my process not right?
P.S.: I am new to the site and it is my first post. Thus, feedbacks on the quality of the post can be helpful.
Thank you.
The magnitude is not quite right. When you square and add, you should get $\sqrt{e^{2t}+2+e^{-2t}}$, which is $e^t+e^{-t}$. Now integration is easy.