AP Calculus AB and BC
Left vs Right Riemann Sum
Left and right Riemann sums differ only in whether each rectangle takes its height from the left or right endpoint of its subinterval. For an increasing function the left sum underestimates and the right sum overestimates, and for a decreasing function the roles swap.
Left Riemann sum
Use when: You are told to use left endpoints, or you want an underestimate for an increasing function.
Right Riemann sum
Use when: You are told to use right endpoints, or you want an overestimate for an increasing function.
Side by side
| Left sum | Right sum | |
|---|---|---|
| Height from | Left endpoint of each subinterval | Right endpoint of each subinterval |
| Skips which data point | The last one | The first one |
| Increasing | Underestimate | Overestimate |
| Decreasing | Overestimate | Underestimate |
Monotonicity, not concavity, decides the direction of the error for these two. Concavity is what governs the trapezoidal and midpoint errors, so mixing the rules up is a common source of wrong justifications.
Averaging the left and right sums gives exactly the trapezoidal rule, which is usually closer to the true value than either one alone.
The mistake
Counting data points as subintervals. A table with values at four times gives three subintervals, and if the spacing is uneven then is not constant and each rectangle needs its own width.
Frequently asked questions
Which sum should I use if the function is not monotonic?
Neither direction can be justified globally. Split the interval where the function changes direction, or use the trapezoidal rule and justify with concavity instead.
In the CED: Unit 6: Integration and Accumulation