Answer
$e-e^{-1}$
Work Step by Step
Given: $r(t)=\sqrt2i+e^tj+e^{-t}k$ ; $0 \leq t \leq 1$
To calculate the length of the curve we will have to use the formula:
$L=\int_a^b |r'(t)| dt$
Thus,
$r'(t)=\lt \sqrt2, e^t, (-e^{-t})\gt$
and $|r'(t)|=\sqrt {(\sqrt2)^2+( e^t)^2+(- e^{-t})^2}dt$
$=\sqrt{ 2+e^{2t}+ e^{-2t}}$
$L=\int_{0}^1 (\sqrt{ 2+e^{2t}+ e^{-2t}})dt=\int_{0}^1 (\sqrt{ (e^{t}+ e^{-t})^2}dt$
$=\int_{0}^1 (e^{t}+ e^{-t})dt$
$L=(e^{t}+ e^{-t})|_{0}^1 =(e^{1}-e^{(0)}+ e^{-1}-e^{0})=e-e^{-1}$
Hence, $L=e-e^{-1}$
