Answer
$29.29$
Work Step by Step
We are given the vector:
$r(t)=\lt t \cos t, t \sin t, 3t \gt$
The vector derivative is:
$r'(t) =\lt -t \sin t+\cos t, t \cos t+\sin t, 3 \gt$
We calculate the length by integration:
$l=\int_{p}^{q}\|r'(t)\|dt=\int_{0}^{2 \pi}\sqrt{t^2+10} dt\\=\dfrac{1}{2}[t \sqrt {t^2+10}+10 \ln (t+\sqrt {t^2+10})]_{0}^{2 \pi}=\pi \sqrt {4\pi^2+10}+5 \ln (2 \pi+\sqrt {4 \pi^2+10})-5 \ln \sqrt{10} \approx 29.29$
