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 1st
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 2nd 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 3rd 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:
