Trapezoidal Approximation Trapezoidal Rule
Error in the Trapezoidal Approximation
Approximation using Parabolas-Simpson's Rule
Error in Simpson's Rule
Which Rule Gives Better Results?
Round-off Errors

                        Numerical Integration

Trapezoidal Rule



The Trapezoidal Rule approximates the area under a curve by using
trapezoids, summing the areas of all the trapezoids.



The are the values of the function    at the partition points
, , , . . . ,,

where is the step size.

See Example 1 - 2, pages 374 - 375.


Error Estimate for The Trapezoidal Rule
If
is continuous and   is any upper bound for the value of 
    on   ,  then
,
where
   is the step size.

  is the error in the Trapezoidal Approximation .

Simpson's Rule


Simpson's Rule approximates the area under a curve by using parabolas,
summing the areas of all the parabolas.



The  are the values of   at the partition points
, , , . . . ,,
The number is even and
   is the step size.

See Example 4, page 378.


Error Estimate for Simpson's Rule
If
 is continuous and  is any upper bound for the value of    
on   , then
                           ,
where
   is the step size.

      is the error in the Simpson Approximation .

See Example 5, page 379.


Which Gives Better Results?
Compare the error estimates for the two approximations
   for the Trapezoidal Rule

    for Simpson's Rule.

The M's mean different things. The factors are different.
Ultimately, we have to look at the geometry of the curve

to see whether trapezoids or parabolas, if either, are going to give the better
result.

See Example 6 - 7, pages 379 - 380.

Round-off Errors
Decreasing the step size    reduces the error in both Approximation Rules
in theory. But in practice it may fail to do so. When    is sufficiently small,
round-off errors in the arithmetic required to evaluate the approximations may
so pile up that the error formulas no longer describe what is really happening.
Reducing   below a certain size can, in fact, make the approximations worst.


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