Estimating with simpson rule

Question

I have a question that is supposed to be very easy in a test: We approximate $\displaystyle \int_0^1 x^2$ with Simpsonrule, and 5 intervals. Choose solution:

Firstly I don't understand what is meant with 5 intervals. Are the intervals not supposed to be even? Secondly, it's an easy test question. How can we get the reply fast without calculating $\frac{h}{3}(y_0+4y_1+y_2)?$

score 1 · Accepted Answer · answered May 10 '19 at 13:33

1

Note that the Simpson Rule gives you the exact answer for polynomials of order less than or equal three.

With $5$ intervals you divide each interval into two equal parts and apply the formula, so actually you are working with $ 10$ intervals.

The error term for the Simpson Rule depends on the fourth derivative of the integrand, therefore in your example the answer is $S=1/3$ regardless of the number of intervals.

answered May 10 '19 at 13:33

Mohammad Riazi-Kermani

68,728

Thank you @Mohammad! I don't understand the last sentence, what has the error to do with the answer $\frac{1}{3}$? – Dovendyr May 14 '19 at 03:19
The exact answer to the integral is $1/3$ and if you don not have any error in Simpson rule, then the Simpson result is also $1/3$ – Mohammad Riazi-Kermani May 14 '19 at 03:55

Estimating with simpson rule

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