Study guide and
2 practice problems
on:
Arc length of a parameterization
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If $\bfx(t)$ is a parameterized curve, and $\bfv = \frac{d \bfx}{dt}$ is its velocity, the length of the curve traced out between $t=a$ and $t=b$ is given by $$s = \int_a^b | \bfv(t)| dt.$$
Related topics
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Practice problems
Find the arc length of the helix $x(t) = \cos t, y(t) = \sin t, z(t) = t$ traced from $t=1$ to $t=2$.
Solution
An ant is on a merry-go-ground that is rotating clockwise at $\omega$ radians per second. Initially, the ant is at $(R,0)$. From the ant's perspective, it walks toward the center with speed $v$. Several snapshots in time are as follows:
Find the parameterization of the path taken by the ant (relative to the ground)
Compute the speed of the ant as a function of $t$. When is it largest?
Set up, but do not evaluate, an integral for the arc length of the path taken by the ant between $t=0$ and when the ant reaches the origin
Solution