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Sketch a “four-leaf clover” and compute its area from 0 to 2 π

by RoRi

Define a four-leaf clover 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 (2 \theta) | \, d \theta &= \frac{1}{4} \int_0^{4 \pi} | \cos \theta| \, d \theta \\ &= \frac{1}{2} \int_0^{2 \pi} | \cos \theta | \, d \theta \\ &= 2, \end{align*}

where for the final step we used the previous exercise (Section 2.11, Exercise #12).

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