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I think I know why but I can't represent this accurately with a graph.

I am supposed to show why Simpson's Rule is so far off for the integral $\int_0^{20} \cos \pi x$

I know that the answer should be zero because it repeats on that interval, 10 up and 10 down evenly.

I know that the antiderivative $\sin \pi x$ will be zero for any 0 or pi value so that evaluates to zero.

When I use Simpson's Rule I get

$\frac{2}{2} (1 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 +1) = 9$

I am pretty certain that the reason the numbers are so far off is because the interval is 2 and that will cover an up and down which Simpson's Rule will overestimate but I can not show this on a graph to equal 9.

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    Yes that is why Simpson's Rule fails in this example.2012-06-06

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I am an idiot and I did the graph wrong. It overestimates because it makes a rectange on each subinterval.