# What is the arc length of #f(x) = ln(x^2) # on #x in [1,3] #?

To do this we need to apply the formula for the length of the curve mentioned in: How do you find the length of a curve using integration?

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To find the arc length of the curve ( f(x) = \ln(x^2) ) on the interval ( x \in [1,3] ), we use the arc length formula:

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

Where ( \frac{dy}{dx} ) represents the derivative of ( f(x) ).

First, find ( \frac{dy}{dx} ):

[ f(x) = \ln(x^2) ] [ f'(x) = \frac{d}{dx} \ln(x^2) = \frac{1}{x^2} \cdot 2x = \frac{2}{x} ]

Now, plug ( f'(x) ) into the arc length formula:

[ L = \int_{1}^{3} \sqrt{1 + \left( \frac{2}{x} \right)^2} , dx ]

[ L = \int_{1}^{3} \sqrt{1 + \frac{4}{x^2}} , dx ]

This integral can be challenging to evaluate directly, so it may be simplest to use numerical methods or a computer algebra system to approximate the value of ( 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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