# What is the arclength of #r=sin(theta^2-(7pi)/6) +2thetacos(theta -(3pi)/8)# on #theta in [(7pi)/8,pi]#?

The arc length

The given polar equation:

#s=int_a^b sqrt((sin(θ^2−(7π)/6)+2θ*cos(θ−(3π)/8))^2+ (2 thetacos(theta^2-(7 pi)/6)+2 cos(theta-(3 pi)/8)-2 theta sin(theta-(3 pi)/8))^2 d theta#

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To find the arc length of the curve described by ( r = \sin(\theta^2 - \frac{7\pi}{6}) + 2\theta \cos(\theta - \frac{3\pi}{8}) ) on the interval ([ \frac{7\pi}{8}, \pi ]), use the formula for arc length:

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

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

First, find ( \frac{{dr}}{{d\theta}} ):

[ \frac{{dr}}{{d\theta}} = 2\theta \cdot \left( \cos(\theta^2 - \frac{7\pi}{6}) \cdot 2\theta - \sin(\theta^2 - \frac{7\pi}{6}) \cdot 2\sin(\theta - \frac{3\pi}{8}) + 2\cos(\theta - \frac{3\pi}{8}) \right) ]

Now, plug ( r ) and ( \frac{{dr}}{{d\theta}} ) into the arc length formula and integrate from ( \frac{7\pi}{8} ) to ( \pi ):

[ L = \int_{\frac{7\pi}{8}}^{\pi} \sqrt{\left(\sin(\theta^2 - \frac{7\pi}{6}) + 2\theta \cos(\theta - \frac{3\pi}{8})\right)^2 + \left(2\theta \cdot \left( \cos(\theta^2 - \frac{7\pi}{6}) \cdot 2\theta - \sin(\theta^2 - \frac{7\pi}{6}) \cdot 2\sin(\theta - \frac{3\pi}{8}) + 2\cos(\theta - \frac{3\pi}{8}) \right)\right)^2} , d\theta ]

Evaluate this integral 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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