Is this how you derive the formula for arc length?
posted by Rob .
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

Yes; that is one way.

in the third step, how did you integrate the right side with no deltavariable?

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