Exercise · ZW-ARC-LENGTH-004 · ★☆☆ Basic
Find the length of the helix \(\mathbf r(t)=\langle\cos t,\sin t,t\rangle\) for \(0\le t\le2\pi\).
Choose one option.
Show answer
- B. \(2\sqrt2\,\pi\)
Explanation
\(|\mathbf r'|=\sqrt{\sin^2t+\cos^2t+1}=\sqrt2\), so \(L=\sqrt2\cdot2\pi=2\sqrt2\,\pi\). It is longer than the circle because it also rises.
Review the concept: Arc length
The arc length of a parametric curve \(\mathbf r(t)\), with \(a\le t\le b\), is the total distance \(L\) traveled along it, measured along the curve itself.
\[L=\int_a^b\lvert\mathbf r'(t)\rvert\,dt\]
More practice problems: arc length
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