# How do you find the arc length of the curve #y = 4 ln((x/4)^(2) - 1)# from [7,8]?

The final answer can be seen here.

The general formula for the arc length is as follows:

Thus, take the derivative and simplify.

Much better. Now we can get rid of that ugly square root.

Then some manipulation to make this evaluation easier...ish.

Looks like we probably have to do Partial Fraction Decomposition on this, unfortunately. Oh well.

Thus, equating it back to the original equation:

Not too bad, actually. Now we have, overall:

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To find the arc length of the curve ( y = 4 \ln\left(\left(\frac{x}{4}\right)^2 - 1\right) ) from ( x = 7 ) to ( x = 8 ), you use the formula for arc length:

[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

where ( a = 7 ) and ( b = 8 ).

First, find ( \frac{dy}{dx} ) by differentiating ( y ) with respect to ( x ). Then, substitute ( \frac{dy}{dx} ) into the formula and integrate from ( x = 7 ) to ( x = 8 ) to find the arc length ( L ).

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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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