How to choose a specific quadrature rule?

[dokken]

For the first set of questions, I would recommend Total number of gauss point - #2 by dokken

@Discoverer

In addition, I try to use the following codes to try to compare the degree of the qudrature rules but the result is the same for degree from 1 to 4, so it seems that I should also change the functionspace where the function f lives ?

Yes, to see an effect of the quadrature choice, you cannot just integrate a CG-1 function, as it is exact with degree=1. If you instead use

x=ufl.SpatialCoordinate(domain)
f = x[0]**2

you should see an effect with degree=1 and all the others.