# What is the arclength of #r=-cos(theta/2-(3pi)/8)/theta # on #theta in [(3pi)/4,(7pi)/4]#?

Arc length

The formula for arc length s in Polar coordinates is

#s=0.757882 units

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To find the arc length of the polar curve ( r = -\frac{\cos\left(\frac{\theta}{2} - \frac{3\pi}{8}\right)}{\theta} ) over the interval ( \left[\frac{3\pi}{4}, \frac{7\pi}{4}\right] ), we can 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 ]

Given ( r = -\frac{\cos\left(\frac{\theta}{2} - \frac{3\pi}{8}\right)}{\theta} ), we need to find ( \frac{dr}{d\theta} ) by taking the derivative with respect to ( \theta ).

After finding ( \frac{dr}{d\theta} ), we substitute ( r ) and ( \frac{dr}{d\theta} ) into the arc length formula and integrate with respect to ( \theta ) over the given interval ( \left[\frac{3\pi}{4}, \frac{7\pi}{4}\right] ). This will give us the arc length of the curve within the specified range of ( \theta ).

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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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