Simpsonís Rule

 

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: