Note that
Now using the formula ( 13.3.2) for on the interval we get,
Thus, we have,
i.e.
(13.3.4) |
where is the average value of the second forward differences.
Recall that in the case of linear function, the second forward differences is zero, hence, the Trapezoidal rule gives exact value of the integral if the integrand is a linear function.
0.0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | |
1.00000 | 1.01005 | 1.04081 | 1.09417 | 1.17351 | 1.28402 | 1.43332 | 1.63231 | 1.89648 | 2.2479 | 2.71828 |
and
Thus,