Calculating Arc Length

So we know we can use Integration to calculate the area under a curve. But what about calculating the arc length of a function’s curve?

Same trick, new application

Ok so we know that Calculus is all about taking tiny, easy to work with, things and using those to solve a harder problem. This is no different. Imagine a function $f(x)=x^2$ . You want to calculate the length from $[2,5]$ . In signature Calculus fashion, we’ll take that length and break it up into tiny bits, before eventually summing them all up. Let’s call the tiny bits $ds$.

We can represent $ds$ as the hypotenuse of a triangle with corresponding sides $dx$ and $dy$. That looks like the below:

The formula is also there. I prefer to use the bottom right connotation/