# How do you find the arc length of the curve #y=lnx# over the interval [1,2]?

Apply the arc length formula.

Arc length is given by:

Rearrange:

Integration is distributive:

Simplify:

Integrate directly:

Reverse the substitution:

Insert the limits of integration:

Rearrange for clarity:

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Length =

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To find the arc length of the curve y = ln(x) over the interval [1,2], you use the arc length formula:

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

For the curve y = ln(x), you find (\frac{dy}{dx}), which is (\frac{1}{x}). Then plug this into the formula and integrate from 1 to 2:

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

Solve the 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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