Numerical integration vs. integrals that are zero

[Thomas Okken]

Trying to get a handle on the behavior of INTEG on the HP-42S, I had it integrate SIN (in DEG mode) from 7 to 367, with ACC = 1, 0.1, 0.01, ..., 1e-15, and charted the results, the estimated absolute error, and the number of points that were sampled:

Code:

    ACC              INTEG                  EPS           PTS

1               -6.63652681894E-1   225.790474144           7
0.1             -6.63652681894E-1    22.5790474144          7
0.01            -7.5600450972E-4      2.2999768795         15
0.001           -7.5600450972E-4      0.22999768795        15
0.0001          -7.5600450972E-4      2.2999768795E-2      15
0.00001          1.872373932E-5       2.29765529718E-3     31
0.000001         3.02829732E-7        2.29202676626E-4     63
0.0000001        1.0434348E-8         2.29202676626E-5     63
0.00000001       3.42144E-11          2.29214420636E-6    127
0.000000001      3.42144E-11          2.29214420636E-7    127
0.0000000001    -6.57E-12             2.29214420636E-8    127
0.00000000001    1.191528E-10         2.29189856856E-9    255
1.E-12           1.194876E-10         2.29189856856E-10   255
1.E-13           1.194876E-10         2.29189856856E-10   255
1.E-14           1.194876E-10         2.29189856856E-10   255
1.E-15           1.194876E-10         2.29189856856E-10   255

I'm not showing the actual sample points; they are distributed as described before by Werner and Thomas K.

There is something odd here: between 1e-6 and 1e-7, the number of points sampled remains the same, 63, but the evaluated integral is different. How can that be? And this happens again between 1e-9 and 1e-10: the same 127 points are sampled, but the results are different. And EPS is clearly also not just a function of the points being sampled.