Hello,
The diagram drawn for this question might be a little misleading because the difference between successive amplitudes is quite large and this might not be typical in a real life system. In a real system usually the difference between successive samples is small. Although not mandatory this helps ensure accuracy.
The reason for the N-1 is probably because you can not use N points when calculating area. If you use N points you get an area that is too big.
For example, if you had N=3 points x1,x2,x3, that only gives you two areas not three, so that's N-1 area segments.
You'll notice that with successive amplitudes that are nearly the same this works out quite well, even if the time increments are quite large. Doing it the other way, with N samples, the error would rise.
I should also mention however that i've seen a definition summing from n1 to n2 and dividing by n2-n1, n1 and n2 inclusive and for a finite interval.