Question

The arc length of the parametric curve: \( x=\cos\theta,\; y=\sin\theta,\; z=\theta \) from \( \theta=0 \) to \( \theta=2\pi \) is equal to _________ (round off to one decimal place).

Hint:

Use the identity \(\sin^2\theta+\cos^2\theta=1\) to simplify arc-length calculations quickly.

Answer 1

Correct Answer: 8.6

The parametric curve is \[ x=\cos\theta,\quad y=\sin\theta,\quad z=\theta. \] Arc length is \[ L=\int_{0}^{2\pi} \sqrt{\left(\frac{dx}{d\theta}\right)^2+\left(\frac{dy}{d\theta}\right)^2+\left(\frac{dz}{d\theta}\right)^2}\, d\theta. \] Compute derivatives: \[ \frac{dx}{d\theta}=-\sin\theta,\quad \frac{dy}{d\theta}=\cos\theta,\quad \frac{dz}{d\theta}=1. \] Thus, \[ \sin^2\theta+\cos^2\theta+1 = 2. \] So, \[ L=\int_{0}^{2\pi}\sqrt{2}\, d\theta = 2\pi\sqrt{2}. \] Numerically, \[ 2\pi\sqrt{2} \approx 8.9. \]