The arc length of logarithm base B has closed form: InputForm[Integrate[Sqrt[1 + 1/(x^2*Log[B]^2)],x]] = (x*Sqrt[1 + 1/(x^2*Log[B]^2)]*(Sqrt[1 + x^2*Log[B]^2] + Log[x] - Log[1 + Sqrt[1 + x^2*Log[B]^2]]))/Sqrt[1 + x^2*Log[B]^2]

arc length common logarithm from 1 to 10 = 9.0834719397922

arc length natural log from 1 to e = 1 - sqrt(2) + sqrt(1 + e^2) + log(1 + sqrt(2)) - log(1 + sqrt(1 + e^2)) = 2.00349711162735247857

arc length log base 2 from 1 to 2 = 1.42116

(via Wolfram Alpha and Mathematica)

Previously, on why we might be interested in the arc length. It turns out arc length is dominated by movement in the x direction, so it is probably not the right measure to use.

## No comments :

Post a Comment