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One way to think about is that these techniques work by integrating exactly the polynomial that interpolates the function where you're sampling it, so you need to resolve the features of the function to get good accuracy.

The Nyquist sampling theorem is of course proved by considering a Fourier transform, which is given by an integral, so the relation to integration in general should not be surprising.

Well in both cases it comes down to aliasing I think (high frequency wave presents as low frequency wave with too big step size)