Given the equation $r=4\cos(3\theta)$, how can I find the domain of each petal? Help!
[Math] How to find the domain $\theta$ for each petal in the Polar graph $r=4\cos(3\theta)$
polar coordinates
Related Solutions
I just added this plot to show what is Raskolnikov trying to tell you. You can find the area. Note the integral he wrote above.
A sketch is useful here, but the only important observation is that $r=0$ when $\theta=0$, and again at $\frac{\pi}{3}$. These are your limits for one petal.
Since the area of a polar curve between the rays $\theta=a$ and $\theta=b$ is given by $\int_{a}^{b}\frac{1}{2}r^{2}d\theta$, we have
$$A=\int_{0}^{\pi/3}\frac{1}{2}\sin^{2}(3\theta)d\theta=\frac{1}{2}\int_{0}^{\pi/3}\frac{1-\cos(6\theta)}{2}d\theta$$
$$=\frac{1}{4}\left[\theta-\frac{\sin(6\theta)}{2}\right]^{\pi/3}_{0}=\frac{1}{4}\left(\frac{\pi}{3}-\frac{1}{2}\sin\left(\frac{6\pi}{3}\right)\right)=\frac{\pi}{12}$$
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Best Answer
This is the polar graph of your function $r=4\cos(3\theta)$:
As mentioned in the comment from @coffeemath; by domain I will assume that you mean the values for $\theta$ to make each petal. So the interval in this case is simply $\cfrac{2\pi}{3}$ for each petal as the full domain ($2\pi$) must be divided equally across each petal. So this means that $-\cfrac{\pi}{6} \le \theta \le \cfrac{\pi}{6}$ is the domain for the first petal (centred on the $x$-axis).