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Sketch a “limacon” and compute its area from 0 to 2 π

by RoRi

Define a limacon by:

    \[ f(\theta) = 2+\cos \theta, \qquad 0 \leq \theta \leq 2 \pi. \]

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

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Then, we compute the area,

    \begin{align*} a(R) = \frac{1}{2} \int_0^{2 \pi} (4 + 4 \cos \theta + \cos^2 \theta) \, d \theta &= 2 \int_0^{2 \pi} d \theta + 2 \int_0^{2 \pi} \cos \theta \, d \theta + \frac{1}{2} \int_0^{2 \pi} \cos^2 \theta \, d \theta \\ &= 4 \pi + 0 + \frac{\pi}{2} \\ &= \frac{9 \pi}{2}. \end{align*}

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