>>13571508>I do not understand the very last sentence. How is this relevant to the discussion?You have to guarantee 1-1 correspondence between the intervals over which you integrate in order to ensure that you calculate the same length regardless of parameterization and this is how you do it. The first interval is the domain which guarantees bijectivity of the reparameterization, . The second interval listed is the image of under , i.e. the image under which if you could calculate the inverse function (you can't do it algebraically in this case), you would guarantee the inverse is bijective and goes back to the same spot. So is the interval you are supposed to use for and is the interval you're supposed to use for in order to guarantee that the two curves are reparameterizations of eachother.
>I can change the intervalYou have to change the intervals in general so that you are actually working with the same curve
>but this would trace out a different curve.Over the interval , both your curves are actually tracing out a different curve (in terms of the behavior of the parameterizations) though they both have the same image set. If you were to draw it by hand for example, the curve starts to "wobble" a little bit when tracing out the curve due to the fact that you have a periodic terms where it doesn't wobble in the case of . To see this wobble, graph out and . Not the same curves, but this helps you see what the first term in each ordered pair does to the curve and you'll notice the wobble in the second parameterization.
https://www.desmos.com/calculator/dbpuqu7qvb 
