There are many applications for the arc length of functions. I came across it when I needed to calculate the length of a density function in an interval . The formula reads

Do you grasp how this formula arises? No? Let me show you!

I imagined how I zoom into a function ย until single points become visisble. Actually, this is not possible from a mathematical point of view – but let’s try for simplicity’s sake ๐ The following sketch shows what I mean:

As you see, the horizontal distance is denoted by while the vertical distance is denoted by . can be expressed in terms of and the derivative :

At the same time, Pythagoras’ theorem holds and allows us to calculate the infinitesimal distance . I also applied some manipulations to the expression.

Adding these endlessly, small sections of the line together is done via a sum. Just write the integral sign in front of the formula.

Done. ๐

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