Answer
$10\sqrt{10}$
Work Step by Step
Given: $r(t)=\lt t, 3 \cos t, 3 \sin t \gt$ ; $-5 \leq t \leq 5$
To calculate the length of the curve we will have to use the formula:
$L=\int_a^b |r'(t)| dt$
Thus,
$r'(t)=\lt 1, -3 \sin t, 3 \cos t \gt$
and $|r'(t)|=\sqrt {1^2+(-3 \sin t)^2+(3 \cos t)^2}dt=\sqrt {1+9 \sin^2 t+9 \cos^2 t}$
$=\sqrt{10}$
$L=\int_{-5}^5 \sqrt{10}dt= \sqrt{10}t|_{-5}^5$
$=5 \sqrt{10}-(-5\sqrt{10})$
Hence, $L=10\sqrt{10}$
