To derive Simpson's Rule and it's error, we can integrate a 2nd degree Lagrange polynomial. To make the process less laborious, we can integrate over the interval [-h,h] using the points (-h,f(-h)), (0,f(0)), (h,f(h)). We end up with,

where denotes the second degree Lagrange polynomial. If I now wish to bound the error I can let and call the error such that,

The integral however, is zero and so I end up with a useless error bound. In one of my books they state a similar thing, but instead of using , they use to get,

I've tried plugging the integral into wolframalpha and get zero here as well. The book though ends up with,

.

Am I doing something wrong here? Are there easier ways of deriving Simpson's along with the error term?Any comments are highly appreciated! Thanks.