Is this how you derive the formula for arc length?

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For a smal change in x, dx:

ds² = dx² + dy²

ds = sqrt [(dx² + dy²)]

s = INTEGRAL of sqrt [(dx² + dy²)]

s = INTEGRAL of sqrt [(dx² + dy² * dx²/dx²)]

s = INTEGRAL of sqrt[(1 + dy² * 1/dx²)] dx

s = INTEGRAL of sqrt[(1 + (dy/dx)²)] dx

• Is this how you derive the formula for arc length? -

Yes; that is one way.

• Is this how you derive the formula for arc length? -

in the third step, how did you integrate the right side with no delta-variable?

• Is this how you derive the formula for arc length? -

There is a delta variable dx. You must compute and insert dy/dx into the integrand to get the resulting arc length

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