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Contents Index Calc Dark Mode Prev Up Next \(\newcommand{\R}{\mathbb{R}}
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Activity 10.3.2 .
In this activity, we will use parameterizations to find the length of some common curves.
(a) Parameterize a circle of radius
\(3\) centered at the origin. Give bounds on the parameter.
(b) Use your parameterization from the previous part in the definite integral of
Theorem 10.3.5 to calculate the circumference of a circle of radius
\(3\text{.}\)
(c) Find the exact length of the spiral defined by
\(\vr(t) = \langle 3\cos(t), 3\sin(t), t \rangle\) on the interval
\([0,2\pi]\text{.}\)
(d) Explain why your result for the length of the spiral is larger than the circumference of the circle of the same radius.
Notice here that while we focused on the circle of radius
\(3\text{,}\) changing to the more general radius of
\(R\) for the circle gives a parameterization of
\(\langle R\cos(t),R\sin(t)\) with
\(0\leq t\leq 2\pi\text{.}\) The work to use the arc length integral to find the circumference of this circle requires only small modifications to the work you did in this activity.
