Arc Length with Vector Functions — Question 1

Question 1

A particle follows the helix $\vec r(t)=\left\langle 3\cos t,3\sin t,4t\right\rangle$ for $0\le t\le 2\pi$. Find the exact distance traveled. Then explain why the vertical rise alone is not the distance.

Question 1 – Solution

Strategy Distance traveled is the integral of speed, not the net change in any one coordinate.

Speed $$\vec r'(t)=\left\langle -3\sin t,3\cos t,4\right\rangle,\qquad \|\vec r'(t)\|=\sqrt{9\sin^2t+9\cos^2t+16}=5.$$

Arc length Therefore $$\boxed{L=\int_0^{2\pi}5\,dt=10\pi}.$$

Interpretation The vertical rise is $4(2\pi)=8\pi$, but the particle simultaneously travels around the cylinder. The velocity has perpendicular horizontal and vertical components of magnitudes $3$ and $4$, producing total speed $5$.