Exercise · ZW-ARC-LENGTH-005 · ★☆☆ Basic
Find the length of the semicircle \(\mathbf r(t)=\langle2\cos t,2\sin t\rangle\), \(0\le t\le\pi\).
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- C. \(2\pi\)
Explanation
\(|\mathbf r'|=|\langle-2\sin t,2\cos t\rangle|=2\), so \(L=\int_0^\pi2\,dt=2\pi\), half of \(2\pi\cdot2\).
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.
\[L=\int_a^b\lvert\mathbf r'(t)\rvert\,dt\]
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