AP Calculus AB and BC glossary

Sample point

Also called: Evaluation point

A sample point is the input inside a subinterval where a Riemann rectangle takes its height. Left, right, and midpoint sums are the three standard choices, but any point in the subinterval is allowed, and for a continuous function every choice gives the same limit.

Written $x_i^*$ , it can be any point in the $i$ th subinterval. For a continuous function the limit $\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x$ is the same number no matter how the points are picked, and that independence is what lets the definite integral be defined at all. Left, right, and midpoint are conventions, not requirements.

The extreme choices bracket the answer. Taking the smallest value of $f$ on each subinterval gives a sum below the integral, taking the largest gives one above, and every other set of sample points lands somewhere between those two.

The mistake

Averaging the two endpoint heights and calling it a midpoint sum. A midpoint sum needs $f$ evaluated at the centre of the subinterval, $f\left(\frac{x_{i-1} + x_i}{2}\right)$ . Averaging $f(x_{i-1})$ and $f(x_i)$ instead builds a trapezoid, which is a different estimate.

Appears in: Unit 6: Integration and Accumulation