# How do you find the length of the polar curve #r=cos^3(theta/3)#?

Use the chain rule.

By Chain Rule,

by cleaning up a bit,

Let us first look at the curve

Note that

Let us now find the length

by pulling

by

I hope that this was helpful.

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To find the length of the polar curve ( r = \cos^3\left(\frac{\theta}{3}\right) ), you can use the arc length formula for polar curves:

[ L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{{dr}}{{d\theta}}\right)^2} , d\theta ]

where ( r ) is the function that defines the curve, and ( \alpha ) and ( \beta ) are the initial and final angles, respectively.

In this case, ( r = \cos^3\left(\frac{\theta}{3}\right) ), so we need to find ( \frac{{dr}}{{d\theta}} ) and then integrate from ( \alpha ) to ( \beta ), the range of ( \theta ).

[ \frac{{dr}}{{d\theta}} = \frac{{d}}{{d\theta}}\left(\cos^3\left(\frac{\theta}{3}\right)\right) ]

Apply the chain rule to differentiate ( \cos^3\left(\frac{\theta}{3}\right) ).

[ \frac{{dr}}{{d\theta}} = -\frac{1}{3}\sin\left(\frac{\theta}{3}\right)\cos^2\left(\frac{\theta}{3}\right) ]

Now, plug ( r ) and ( \frac{{dr}}{{d\theta}} ) into the arc length formula and integrate from the initial angle ( \alpha ) to the final angle ( \beta ).

[ L = \int_{\alpha}^{\beta} \sqrt{\cos^6\left(\frac{\theta}{3}\right) + \frac{1}{9}\sin^2\left(\frac{\theta}{3}\right)\cos^4\left(\frac{\theta}{3}\right)} , d\theta ]

After integration, you will get the length of the polar curve within the given range of angles.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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