What is the arclength of the polar curve #f(theta) = 5costheta+sin6theta # over #theta in [0,pi/3] #?
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To find the arc length of the polar curve (f(\theta) = 5\cos(\theta) + \sin(6\theta)) over the interval (\theta \in [0, \frac{\pi}{3}]):
 Compute the derivative of (f(\theta)) with respect to (\theta).
 Square the derivative and add 1.
 Integrate the square root of the expression obtained in step 2 over the interval (\theta \in [0, \frac{\pi}{3}]).
 Evaluate the integral to find the arc length.
Let's go through the steps:

The derivative of (f(\theta)) is (f'(\theta) = 5\sin(\theta) + 6\cos(6\theta)).

Square the derivative and add 1: (f'(\theta)^2 + 1 = (5\sin(\theta) + 6\cos(6\theta))^2 + 1).

Integrate (\sqrt{f'(\theta)^2 + 1}) over the interval (\theta \in [0, \frac{\pi}{3}]): [\text{Arc Length} = \int_{0}^{\frac{\pi}{3}} \sqrt{(5\sin(\theta) + 6\cos(6\theta))^2 + 1} , d\theta]

Evaluate the integral to find the arc length over the given interval.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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