Integrate the fifth degree polynomial (x + 2*y + 1)^5 over
the domains triangle with hole and annulus
which your worked with in the meshing project.
Here is what I am getting:
./triangle-with-hole Usage: ./triangle-with-hole d a d = integration strength a = maximal triangle area
./triangle-with-hole 5 0.1 requested integration strength = 5 used integration strength = 5 domain is a triangle with hole vertices = 31, edges = 65, elems = 34 expected integral = 463.1075439453125 calculated integral = 463.1075439 error = -2.28511e-11
./annulus Usage: ./annulus d a n d = integration strength a = maximal triangle area n = number of outer edges on the annulus
./annulus 5 0.1 24 requested integration strength = 5 used integration strength = 5 domain is an annulus (really a 24-gon) of radii 0.325 and 0.65 vertices = 74, edges = 174, elems = 100 exact integral = 11.21156654 calculated integral = 10.92885656 error = 0.28271
./annulus 5 0.1 48 requested integration strength = 5 used integration strength = 5 domain is an annulus (really a 48-gon) of radii 0.325 and 0.65 vertices = 219, edges = 561, elems = 342 exact integral = 11.21156654 calculated integral = 11.14019861 error = 0.0713679
The "exact answer" in those outputs is the value of the integral over a true (round) annulus. We are approximating the annulus with a polygon, so some of the error is due to that approximation.
Experiment with your code to confirm that every doubling the number of sides cuts down the error by a factor of four. I think that's not difficult to prove but I haven't done it.