Simpson’s Rules

Basic assumption:

The curved portion of a figure forms part of a parabola

y=a₀+a₁x+a₂x²+a₃x³

and gives the area contained between 3 consecutive equally spaced ordinates.

Simpson’s  1^st rule

-is used to find area when there are odd number of evenly spaced ordinates.

- This is called “3 ordinate rule”

This rule can be generalized to any figure defined by an odd number of evenly spaced ordinates, by applying the First Rule to ordinates 0 to 2, 2 to 4, 4 to 6 and so on, and then summing the resulting answers.

This provides the rule for (n + 1) ordinates:

Simpson’s  2^nd  rule

-is used to find area when there are even number of evenly spaced ordinates.

The rule for four evenly spaced ordinates is:

It can be extended to cover 7, 10,13, etc., ordinates, becoming:

Simpson’s  3^rd  rule

-is a special case to find area between any 2 ordinates when 3 evenly spaced ordinates are known.

- This is called Simpson's 5, 8 minus 1 Rule.

Unlike other Simpson's rules, the 5, 8, -1 rule cannot be applied to moments.

A corresponding rule for moments, derived in the same way as those for areas,

is known as Simpson's 3, 10 minus 1 rule and gives the moment of the area bounded by y₀ & y₁ about y₀, as: