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Sketch a “four-leaved rose” and compute its area from 0 to 2π

by RoRi

Define a four-leaved rose by:

    \[ f(\theta) = |\sin (2 \theta)|, \qquad 0 \leq \theta \leq 2 \pi. \]

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} \sin^2 (2 \theta) \, d \theta &= \frac{1}{4} \int_0^{4 \pi} \sin^2 \theta \, d \theta \\ &= \frac{1}{8} \int_0^{4 \pi} d \theta - \frac{1}{8} \int_0^{4 \pi} \cos (2 \theta) \, d \theta \\ &= \frac{\pi}{2} - \frac{1}{16} \int_0^{8 \pi} \cos \theta \, d \theta \\ &= \frac{\pi}{2}. \end{align*}

One comment

  1. Kaustubh Bansal says:

    Your figures look really neat. Are you using Latex? Would you ming sharing your latex source for the figure here?

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