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

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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}{2} \right] ), use the formula for arc length in polar coordinates:

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

where ( r = f(\theta) ) and ( \frac{dr}{d\theta} ) is the derivative of ( f(\theta) ) with respect to ( \theta ).

Compute ( \frac{dr}{d\theta} ) by differentiating ( f(\theta) ) with respect to ( \theta ), then plug it into the formula along with ( r = f(\theta) ).

Evaluate the integral over the given interval ( \left[ \frac{\pi}{12}, \frac{\pi}{2} \right] ).

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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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