Exercise · ZW-ARC-LENGTH-006 · ★★☆ Intermediate
For \(\mathbf r(t)=\langle e^t\cos t,\,e^t\sin t\rangle\), with \(0\le t\le1\), what is \(L\)?
Choose one option.
Show answer
- A. \(\sqrt2\,(e-1)\)
Explanation
One finds \(|\mathbf r'(t)|=\sqrt2\,e^t\), so \(L=\sqrt2\int_0^1e^t\,dt=\sqrt2\,(e-1)\).
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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