Calculus II · Parametric and polar curves
Separate the two loops before integrating the limacon
Problem
For the polar curve \[ r=1-2\sin\theta,\qquad 0\le\theta\le2\pi, \] find the parameter values where the curve passes through the pole. Explain which interval traces the inner loop and where that loop lies geometrically despite $r<0$. Compute the exact inner-loop area, total area swept over $0\le\theta\le2\pi$, and outer-loop area.
Hint
Solve $1-2\sin\theta=0$.
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