Question

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

Answer 1

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 
     $\:\int _{\frac{\pi }{3}}^{\pi }\:\sqrt{\left(4+2cos\left(3\theta \right)\right)^2+\left(-6sin\left(3\theta \right)\right)^2}d\theta =12.0999$
     

Answer 2

The length of each petal of the polar curve is 12.099.

We have given that  the polar curve r=4+2cos3 theta

What is the formula for arc length in polar form?

$s=\sqrt{r^2+\frac{dr}{d\theta}^2 }d\theta$

   

Differentiate r with respect to θ we get,

$\frac{d}{dx}(4+2 cos(3x)=-6sin(3x)$

   

Therefore,the arc length of 1 petal is,

Integral(pi/3 to pi )$\sqrt{(4+2cos (3\theta))^2+(-6sin (3\theta))^2d\theta} =12.099$

Therefore, the length of each petal of the polar curve is 12.099.

To learn more about the arc length visit:

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