# What is the antiderivative of the distance function?

Jun 29, 2015

The distance function is:

$D = \sqrt{{\left(\Delta x\right)}^{2} + {\left(\Delta y\right)}^{2}}$

Let's manipulate this.

$= \sqrt{{\left(\Delta x\right)}^{2} + {\left(\Delta y\right)}^{2} / {\left(\Delta x\right)}^{2} {\left(\Delta x\right)}^{2}}$

$= \sqrt{1 + {\left(\Delta y\right)}^{2} / {\left(\Delta x\right)}^{2}} \Delta x$

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

$= \sum \sqrt{1 + {\left(\Delta y\right)}^{2} / {\left(\Delta x\right)}^{2}} \Delta x$

$= \int \sqrt{1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2}} \mathrm{dx}$

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