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 $\frac{h}{3}(y_0+4y_1+y_2)?$
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.