Given a definite integral $\int_a^bf$,
If we increase the number of nodes and weights of the Gaussian quadrature, we would get closer to the exact integral.
But another way to get more exact is to partition the interval $[a,b]$ into many nodes, and add as done in the Riemann sum.
For both the midpoint (which happens to be the 1 point Gaussian) and the trapezoid quadratures, as we double the number of nodes, the error decreases by $\sim 1/4$. I have learned that this could be written as $\mathcal{O}(h^2)$, or order 2 accuracy.
My question is, if we fix the quadrature (say, 3 point Gaussian) then what is the order of accuracy? I am aware of the theorem that says the Gaussian quadrature is accurate for polynomials of order $2n-1$. Does this help us in any way?
Another thing that I noticed was that for $\sin(x)$, the piecewise 3 point Gaussian quadrature had better accuracy than for the 4 point Gaussian. Does this have anything to do with the fact that sine is an odd function?