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Sketch a “rose petal” and compute its area from 0 to π/2

by RoRi

Define a rose petal by:

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

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

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