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I do not know what this questions is asking for: I know how to solve problems with trapezoidal and Simpson's rule. But I dont know what this question wants. Any help please?

Estimate the minimum number of subintervals needed to approx. the integral $$\int_0^1 x \, dx.$$

with an error of magnitude less than $10^{-4}$ by a) trapezoidal b) Simpson's?

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  • $\begingroup$ We are assuming $N$ equally spaced intervals? $\endgroup$ – Tyler Hilton Apr 12 '15 at 7:09
  • $\begingroup$ The trapezoidal rule and Simpson's rule come with error estimates, and wherever you learned about those rules there was probably a section on those error terms. That's what you need to use. If all else fails, Google. $\endgroup$ – Gerry Myerson Apr 12 '15 at 7:14
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HINT:

The question is:

a) find the $n$-th trapizoidal term $T_n$ such that $$\Big|\int_0^1f(x)\ dx-T_n\Big|<10^{-4}$$ - Try googeling: error estimates for the trapezoid rule.

b) find the $n$-th Simpson's term $S_n$ such that $$\Big|\int_0^1f(x)\ dx-S_n\Big|<10^{-4}$$ - Try googeling: error estimates for the Simpson's rule.

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Since the integrand is linear, the trapezoidal rule will give the exact answer for any number of subintervals; so $n=1$ would be the minimum number of subintervals required.

For the same reason, Simpson's rule will give the exact answer for any partition of $[0,1]$ into an even number of subintervals, so $n=2$ would be the minimum number of subintervals needed.


Notice in general that the error estimate for the trapezoidal rule involves the maximum value of $\left|f^{\prime\prime}(x)\right|$, and the error estimate for Simpson's rule involves the maximum of $\left|f^{(4)}(x)\right|$.

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