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Simpson's Rule Notes- By Popular Demand
Here is a rough outline on how to solve a simpson's rule problem:
Prerequisite: Take all the derivatives up to and including the fourth
derivative of the function you are integrating.
1. Determine "n" from the error formula: (abs(max
f''''(c))(b-a)^5)/(180n^4), by setting this term less than or equal to the
prescribed error. Note that the abs(max f''''(c)) term is maximum value of
the fourth derivative over the interval [a,b] that you are integrating over.
Get the maximum value by plugging the fourth derivative in your graphing
calculator and looking at your function.
(Note:This includes the endpoints! Do not forget to look at them.) .
Once you have solved for "n", round UP to
the nearest EVEN integer. e.g. if n>=6.234 round up to n=8 .
2. Once you have the value for "n" you can determine the value "h", which is
the "height" of each "rectangle" you are breaking up the region into. h=
(b-a)/n, where a and b are the beginnning and endpoints of the integral
respectively, and n is what you solved for in step 1.
3. Put this all together to compute the estimate of the integral to within
the prescribed error: integral from a to b of f(x) is approximately:
[(h/3)[f(a)+4f(a+h)+2f(a+2h)+4f(a+3h)+2f(a+4h)+..........+4f(a+(n-1)h)+f(b)]].
Note that you keep adding the h increment to the function until you get from
f(a) to f(b). Keep alternating, starting with 4f(..) then 2f(...) then
4f(...) after you leave f(a) until you "hit" f(b). The first term after f(a)
is always 4f(...) and the term right before f(b) is always 4f(...). Also
notice that in total, you will need to evaluate the function a total of n+1
times to come up with the estimate.
Russ
