There are several natural systematic choices for here, which we will name:

They correspond to choosing the leftmost point in each interval, the rightmost point, the point with largest and with smallest value for the integrand.

We denote them by L, R, T, B respectively.

They are defined explicitly by:

1. If is the left endpoint,

2. If is the right endpoint,

3. Let m_k be the smallest value of f in the kth interval and M_k the largest.

We then have:

and

Comment 1

Comment 2

Sums and integrals

Illustrations: n = 5, 10, 25, 50 infinite Formulae