Find the length of each petal of the polar curve

Ответ:

benji2246

13.03.2022

The graph of the given polar curve is shown in the figure. One of the petals traverses from π/3 to π.

The formula in solving for the arc length in polar form is
$s=\sqrt{r^2+\left(\frac{dr}{d\theta }\right)^2}d\theta$

The derivative of r with respect to θ is
$\frac{d}{dx}\left(4+2\cos \left(3x\right)\right)=-6\sin \left(3x\right)$

So, the arc length of 1 petal is
$\:\int _{\frac{\pi }{3}}^{\pi }\:\sqrt{\left(4+2cos\left(3\theta \right)\right)^2+\left(-6sin\left(3\theta \right)\right)^2}d\theta =12.0999$

Find the length of each petal of the polar curve r=4+2cos3 theta