Answer
$1.2780$
Work Step by Step
Given: $r(t)=\lt \sin t, \cos t, \tan t\gt$; $0 \leq t \leq \frac{\pi}{4}$
To calculate the length of the curve we will have to use the formula:
$L=\int_a^b |r'(t)| dt$
Thus,
$r'(t)=\lt \cos t, -\sin t, \sec^2 t\gt$
and $|r'(t)|=\sqrt {( \cos t)^2+(-\sin t)^2+(\sec^2 t)^2}dt$
$=\sqrt{ 1+\sec^4t}$
$L=\int_{0}^\frac{\pi}{4}(\sqrt{ 1+\sec^4t}) dt$
As per question, we will use calculator to find the length of the curve.
Hence, $L= 1.2780$
