Arc Length with Vector Functions — Question 3

Question 3

For $\vec r(t)=\left\langle t^2,\frac 23t^3,0\right\rangle$, find the arc length from the point corresponding to $t=1$ to the point corresponding to $t=2$. Also find the chord length and compare the two quantities.

Question 3 – Solution

Strategy Use the same speed structure as in Question 2, but retain the nonzero lower limit.

Arc length $$L=\int_1^2 2t\sqrt{1+t^2}\,dt =\boxed{\frac 23(5\sqrt 5-2\sqrt 2)}.$$

Chord The endpoints are $P=(1,2/3,0)$ and $Q=(4,16/3,0)$, so $$\|Q-P\|=\sqrt{3^2+\left(\frac{14}{3}\right)^2} =\boxed{\frac{\sqrt{277}}3}.$$

Comparison Numerically, $L\approx 5.568$ while the chord is about $5.548$. The curve length is slightly larger, as required; equality would occur only for a straight segment traversed without reversal.