What is the antiderivative of the distance function?

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Answer 1 · 2015-06-29T07:02:07

The distance function is:

$D = sqrt((Deltax)^2 + (Deltay)^2)$

Let's manipulate this.

$= sqrt((Deltax)^2 + (Deltay)^2/(Deltax)^2 (Deltax)^2)$

$= sqrt(1 + (Deltay)^2/(Deltax)^2)Deltax$

Since the antiderivative is basically an indefinite integral, this becomes an infinite sum of infinitely small $dx$ :

$= sumsqrt(1 + (Deltay)^2/(Deltax)^2)Deltax$

$= int sqrt(1 + ((dy)/(dx))^2)dx$

which happens to be the formula for the arc length of any function you can manageably integrate after the manipulation.