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

by RoRi

Define a lazy eight by:

    \[ f(\theta) = \sqrt{|\cos \theta|}, \qquad 0 \leq \theta \leq 2 \pi. \]

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

The sketch is as follows:

Rendered by QuickLaTeX.com

Then, we compute the area,

    \begin{align*} a(R) = \frac{1}{2} \int_0^{2 \pi} | \cos \theta | \, d \theta &= \frac{1}{2} \left( \int_0^{\frac{\pi}{2}} \cos \theta \, d \theta - \int_{\frac{\pi}{2}}^{\frac{3 \pi}{2}} \cos \theta \, d \theta + \int_{\frac{3 \pi}{2}}^{2 \pi} \cos \theta \, d\theta \right) \\ &= \frac{1}{2} (1 - (-1) + 1 - (-1)) \\ &= 2. \end{align*}

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