Hello, I am having trouble remembering some of the material required for my current calculus course so I am reviewing some of the previous material that I have forgotten. I am having trouble following the definition of The Arc Length Function as presented in James Stewart's "Calculus: Fourth Ed." page 579. I already follow how to derive the formula for arc length. But I am having problems with the concept of the arc length function. I am given the following: Now I am not following the purpose of this section on arc length function. I can already figure out (5) with ds being equal to (3). So what is the purpose of this section? And I am also not following what the text is doing at the beginning by replacing the variable of integration with t and then differentiating and re-replacing t with x again. What the hey? I would really like to know what they are doing in this step. Any help is appreciated. Thankyou.