The generic form of a parabola is _{}.

The area of the parabolic region from _{} to _{} is

_{}

Find _{}, _{}, and _{} by substituting the *x*-values
at those points

_{}

Add the _{} and _{}

_{}

Substituting into the area equation above gives

_{}

So, the area of the parabolic region can be found by adding
first *y*-value, _{}, four times the middle *y*-value, _{}, and the last *y*-value, _{}, then multiplying that sum by one-third of the interval
width, *h*.

Thus, given an odd number of *x*-values, _{}, the area of the region can be estimated by

_{}

Factor out the _{}, replace *h* with _{}, and combine like terms to get Simpson’s Rule:

_{}