Exercise · ZW-ARC-LENGTH-007 · ★★☆ Intermediate
Find the length of \(\mathbf r(t)=\langle t,\tfrac23t^{3/2}\rangle\) for \(0\le t\le3\).
Choose one option.
Show answer
- D. \(\tfrac{14}{3}\)
Explanation
\(\mathbf r'=\langle1,t^{1/2}\rangle\), with \(|\mathbf r'|=\sqrt{1+t}\). So \(L=\int_0^3\sqrt{1+t}\,dt=\tfrac23(4^{3/2}-1)=\tfrac{14}{3}\).
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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