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 Simpsonís Rule: