The figure above shows the graphs of the polar curves \(r=2\cos{(3\theta)}\) and \(r=2\). What is the sum of the areas of the shaded regions?

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The area of a bounded region is given by the integral:

$$ \frac{1}{2}\int \limits_a^b (r(\theta))^2   d\theta $$

A single petal is traced from \(-\dfrac{\pi}{6}\) to \(\dfrac{\pi}{6}\). The area of three petals is give by:

$$ 3\cdot \frac{1}{2}\int \limits_{-\pi/6}^{\pi/6} (2\cos{3\theta})^2   d\theta $$

We can subtract this from the area of the circle to obtain the shaded region:

$$ \pi(2)^2 - \frac{3}{2}\int \limits_{-\pi/6}^{\pi/6} (2\cos{3\theta})^2   d\theta $$ $$ \approx \boxed{9.425}$$