Yeah, that is what I was doing, if you look careful... the post was a joke (and you do not need abs in expansion,you can directly use x and -x, or multiply the intervals with -1 and use two integrals on 0 to 1). However I do not understand this part about final division by 2, that you, uau, and b2 are talking about. Why should you divide the interval? The mean is for ALL the domain, not only for pieces of domain. That is the reason of it, if your interval is longer, your approximation should be for all the interval, not per unit of interval. Otherwise the MSE makes no sense. In this light, their example has a MSE of 0.33333, that's it. They have a typo either in the value or in the function chosen, or used a wrong tool to calculate it. Which is also confirmed by excel (except that the excel shows the "fitness" function of the trendlines, R^2, which is the complementary of MSE, it shows how well the two graphics fit together)

By the way, is this a puzzle, or what? A polynomial approximation is ensured by

Weierstrass theorem and for the 4th degree you have exactly 14 operations to write the full polynomial. Put it in excel, make a graphic, add a polynomial trendline of degree 4 or 6, and you are 0.00013 far away, the rest may be just a bit of optimization...