Question

What is the antiderivative of the distance function?

Answer 1

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.