# What is the arclength of #r=5/2sin(theta/8+(3pi)/16) -theta/4# on #theta in [(-3pi)/16,(7pi)/16]#?

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To find the arc length of the curve described by the polar equation ( r = \frac{5}{2} \sin\left(\frac{\theta}{8} + \frac{3\pi}{16}\right) - \frac{\theta}{4} ) on the interval ( \theta ) in ( \left[-\frac{3\pi}{16}, \frac{7\pi}{16}\right] ), you can use the formula for arc length in polar coordinates:

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

where ( r ) is the given polar function and ( \frac{dr}{d\theta} ) is its derivative with respect to ( \theta ).

Plugging in the given polar function and its derivative, you can evaluate the integral over the specified interval to find the arc length.

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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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