Accurate estimation of arclength

To show the basic principle with a simple example, let's start with x^2 from 0 to 2:

One could use the arclength formula to find the arclength of course:

But an alternative I thought of was to use polynomial interpolation on the arclength f ( square root of 1 plus the derivative squared) and integrate to get:

The reason to use this instead is if you consider more complex curves like:

If you tell Maple to figure this arclength from [-3..1] it will just sit there forever churning unable to do it. So normally you would have to do a kind of sum of straight line approximations along the curve, but If you break the curve into manageable pieces you can use this method I'm proposing. I'll choose the interval [-3..-1] because it's always increasing there, f is square root of 1 plus the derivative of the above function squared ...

I found that 5 point polynomial interpolation is good for getting very close to the correct answer while being very fast to calculate, but if more accuracy was needed or more speed the number of points in the interpolation could be varied. 

**Edit***

I think another approach would be to just break the curve into a certain number of equal parts and then use 3 point interpolation on those pieces, which would be a quadratic Simpson's rule which leads to coefficients like this:

You would sum this over the interval for a certain d diminishing for more accuracy, but apparently never 0 because d is in the denominator.