Exercise · ZW-ARC-LENGTH-009 · ★★★ Advanced
What is the length of \(\mathbf r(t)=\langle\cos2t,\sin2t\rangle\) for \(0\le t\le\pi\), which goes around the unit circle once?
Choose one option.
Show answer
- B. \(2\pi\)
Explanation
Here \(|\mathbf r'|=2\), so \(L=\int_0^\pi2\,dt=2\pi\). It is the same length as with \(\langle\cos t,\sin t\rangle\): length does not depend on how fast the curve is traversed, only on the curve and how many times it is traversed.
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
- ZW-ARC-LENGTH-001 Find the length of mathbf r(t)=langle3t,4trangle for 0≤ t≤2.
- ZW-ARC-LENGTH-002 What is the length of mathbf r(t)=langlecos t,sin trangle for 0≤ t≤2π?
- ZW-ARC-LENGTH-003 Find the length of mathbf r(t)=langle t,2t,2trangle for 0≤ t≤3.
- ZW-ARC-LENGTH-004 Find the length of the helix mathbf r(t)=langlecos t,sin t,trangle for 0≤ t≤2π.
- ZW-ARC-LENGTH-005 Find the length of the semicircle mathbf r(t)=langle2cos t,2sin trangle, 0≤ t≤π.
- ZW-ARC-LENGTH-006 For mathbf r(t)=langle e^tcos t, e^tsin trangle, with 0≤ t≤1, what is L?
- ZW-ARC-LENGTH-007 Find the length of mathbf r(t)=langle t,tfrac23t^3/2rangle for 0≤ t≤3.
- ZW-ARC-LENGTH-008 For mathbf r(t)=langle t^2,t^3rangle, 0≤ t≤1, what is the length?