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Subsections
The purpose of this lab is to give you some experience with using the
trapezoidal rule and Simpson's rule to approximate integrals.
In class we have talked about the trapezoidal rule and Simpson's rule
for approximating the definite integral
Both methods start by dividing the interval [a,b] into n
subintervals of equal length by choosing a partition
satisfying
where
is the length of each subinterval. For the trapezoidal rule, the
integral over each subinterval is approximated by the area of a
trapezoid. This gives the
following approximation to the integral
There is also an error term associated with the trapezoidal rule that
can be used to estimate the error. More precisely, we have
where
for some value c between a and b.
One way to use this error term is as a way to bound the number of
subintervals required to achieve a certain tolerance. That is, suppose
is a small number and we want to determine a value of
n that guarantees
If we substitute the error formula from above into this inequality and
rearrange it to isolate n2 we get the following.
Now, if we let M be the maximum of on the interval
[a,b], we can take the square root of both sides of the equation to
obtain the following estimate for n.
The way to think about this result is that it gives a value for n
which guarantees that the error of the trapezoidal rule is less than
the tolerance . It is generally a very conservative
result. As you will discover in the exercises, the actual number of
subintervals required to satisfy the tolerance is usually much
smaller than the number given by the error estimate.
For Simpson's rule, the function is approximated by a parabola over
pairs of subintervals. When the areas under the parabolas are computed
and summed up, the result is the following approximation.
As for the trapezoidal rule, there is an error formula which says that
where
for some value c between a and b.
As we did for the trapezoidal rule, we can rearrange this formula to
allow us to estimate the number of subintervals required so that we
can guarantee
Using essentially the same steps as we used for the trapezoidal rule, we
get the following inequality.
where M is the maximum of on the interval [a,b].
- 1.
- The Getting started worksheet contains two proofs of
the following result,
which is Simpson's rule applied to the interval [m-h,m+h] using two
subintervals. Explain why this
result implies the formula given below for Simpson's rule over n
subintervals, where n is even.
- 2.
- For the following functions and intervals, complete the
following steps.
- (i)
- By using Maple's int and, possibly, evalf
commands, find a good
approximation to the integral of the function over the given interval.
- (ii)
- Find the minimum number of subintervals required so that Tn
satisfies the tolerance
Compare this number to the one you would get by using the error
estimate for the trapezoidal rule.
- (iii)
- Find the minimum number of subintervals required so that Sn
satisfies the tolerance
Compare this number to the one you would get by using the error
estimate for Simpson's rule.
- (a)
- f(x) = x/(1+x3), interval [0,5].
- (b)
- , interval [0,10].
- 3.
- Consider the error function
Use Simpson's rule to approximate to within an
accuracy of 0.1 and determine the minimum number of subintervals
required. In a previous lab, you used Taylor polynomials to
approximate this same integral. Which method do you think is better?
Justify your answer.
- 4.
- Consider the function
Using Simpson's rule, find the minimum number of steps required to
approximate each of the two integrals below to within 0.001.
and
Can
you use the error estimate to explain why there is such a big
difference between the number of subintervals required?
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Jane E Bouchard
2/1/2000