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.
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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