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What is the arclength of #r=-sin(-5theta^2+(11pi)/8) +3thetacos(7theta -(5pi)/8)# on #theta in [(7pi)/8,(13pi)/8]#?

Answer 1

Charles Arocha

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Answer 2

Emilia Aguilar

To find the arc length of the curve described by (r = -\sin\left(-5\theta^2 + \frac{11\pi}{8}\right) + 3\theta \cos\left(7\theta - \frac{5\pi}{8}\right)) on the interval (\left[\frac{7\pi}{8}, \frac{13\pi}{8}\right]), you use the formula for arc length in polar coordinates. The formula is:

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

First, you calculate (\frac{dr}{d\theta}) and then plug it into the formula along with (r), and integrate from (\frac{7\pi}{8}) to (\frac{13\pi}{8}). This will give you the arc length of the curve over the specified interval.

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Answer from HIX Tutor

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.