Left, right, and midpoint sums

The three standard choices of sample point within each Riemann-sum rectangle.

A left sum samples each rectangle's height at its subinterval's LEFT endpoint; a right sum samples at the RIGHT endpoint; a midpoint sum samples at the subinterval's midpoint. For an increasing function, the left sum underestimates and the right sum overestimates the true area (or vice versa for a decreasing function) — the midpoint sum typically lands closer to the exact value than either endpoint choice.

Examples

For $f(x)=x^2$ (increasing on $[0,b]$ for $b>0$ ), a left sum always underestimates the true area, a right sum always overestimates it.