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Section 9.9 : Arc Length with Polar Coordinates
3. Set up, but do not evaluate, and integral that gives the length of the following polar curve. You may assume that the curve traces out exactly once for the given range of \(\theta \).
\[r = \cos \left( {2\theta } \right) + \sin \left( {3\theta } \right), \,\, 0 \le \theta \le 2\pi \]Show All Steps Hide All Steps
Start SolutionThe first thing we’ll need here is the following derivative.
\[\frac{{dr}}{{d\theta }} = - 2\sin \left( {2\theta } \right) + 3\cos \left( {3\theta } \right)\] Show Step 2We’ll need the \(ds\) for this problem.
\[ds = \sqrt {{{\left[ {\cos \left( {2\theta } \right) + \sin \left( {3\theta } \right)} \right]}^2} + {{\left[ { - 2\sin \left( {2\theta } \right) + 3\cos \left( {3\theta } \right)} \right]}^2}} \,d\theta \] Show Step 3The integral for the arc length is then,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{L = \int_{0}^{{2\pi }}{{\sqrt {{{\left[ {\cos \left( {2\theta } \right) + \sin \left( {3\theta } \right)} \right]}^2} + {{\left[ { - 2\sin \left( {2\theta } \right) + 3\cos \left( {3\theta } \right)} \right]}^2}} \,d\theta }}}}\]