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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

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 valueAs 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- 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
*T*_{n}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
*S*_{n}satisfies the tolerance Compare this number to the one you would get by using the error estimate for Simpson's rule.

- (a)
- , interval [0,4].
- (b)
*f*(*x*) = (2-*x*)^{-1/3}, interval [1,1.99].

- 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 approximation 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?
- 5.
- Simpson's rule usually requires fewer subintervals than the
trapezoidal rule for the same accuracy, but this is not always
true. Consider the following integral.
For this integral, you should find that with an equal number of
subintervals, the trapezoidal rule gives a more accurate
answer. Verify that this is true for several values of
*n*, . Can you explain this by using the error estimates?

11/12/1999