# What is the arclength of the polar curve #f(theta) = sin4theta-cos3theta # over #theta in [pi/3,pi/2] #?

The arclength is approximately

The equation for arclength of a polar curve is:

In this case,

So, the arclength can be found by:

Final Answer

I am not entirely sure how to solve this integral manually; if anyone else has a valid method of integration, feel free to extend this answer, but as far as I can see, the only way to solve this problem is with a graphing calculator or similar interface.

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The arc length of the polar curve ( f(\theta) = \sin(4\theta) - \cos(3\theta) ) over the interval ( \theta \in [\frac{\pi}{3}, \frac{\pi}{2}] ) can be calculated using the formula for arc length in polar coordinates:

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

where ( r = f(\theta) ) is the polar function and ( \alpha ) and ( \beta ) are the limits of integration. Compute ( \frac{dr}{d\theta} ) by differentiating ( r ) with respect to ( \theta ), and then evaluate the integral over the given interval.

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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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