# What is the arclength of the polar curve #f(theta) = 2thetasin(5theta)-thetacot2theta # over #theta in [pi/12,pi/3] #?

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To find the arc length of the polar curve ( f(\theta) = 2\theta \sin(5\theta) - \theta \cot(2\theta) ) over ( \theta ) in ( \left[\frac{\pi}{12}, \frac{\pi}{3}\right] ), we use the formula for arc length of a polar curve:

[ L = \int_{\theta_1}^{\theta_2} \sqrt{[r(\theta)]^2 + \left(\frac{dr}{d\theta}\right)^2} , d\theta ]

where ( r(\theta) ) is the polar curve function and ( \frac{dr}{d\theta} ) is its derivative with respect to ( \theta ).

Here, ( r(\theta) = 2\theta \sin(5\theta) - \theta \cot(2\theta) ).

Taking the derivative:

[ \frac{dr}{d\theta} = 2\sin(5\theta) + 10\theta \cos(5\theta) - \frac{\cot(2\theta)}{2} + \theta\csc^2(2\theta) ]

Now, we plug ( r(\theta) ) and ( \frac{dr}{d\theta} ) into the arc length formula and integrate over the given interval:

[ L = \int_{\frac{\pi}{12}}^{\frac{\pi}{3}} \sqrt{\left(2\theta \sin(5\theta) - \theta \cot(2\theta)\right)^2 + \left(2\sin(5\theta) + 10\theta \cos(5\theta) - \frac{\cot(2\theta)}{2} + \theta\csc^2(2\theta)\right)^2} , d\theta ]

This integral can be evaluated numerically using computational tools like calculators or software.

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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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