AP Calculus AB and BC
Riemann Sum vs Trapezoidal Rule
A Riemann sum approximates with rectangles, using one function value per subinterval, and whether it over or underestimates depends on whether the function is increasing or decreasing. The trapezoidal rule uses both endpoints to form a trapezoid, and its error direction depends on concavity instead.
Riemann sum
Use when: The question names left, right, or midpoint, or you are building the definition of the definite integral.
Trapezoidal rule
Use when: You have a table of values and want a better estimate, or the question asks specifically for trapezoids.
Side by side
| Riemann sum | Trapezoidal rule | |
|---|---|---|
| Shape used | Rectangle | Trapezoid |
| Values per subinterval | One | Both endpoints |
| Error direction set by | Increasing or decreasing | Concavity |
| Overestimates when | Right sum and increasing | Concave up |
| Relationship | Left and right sums bracket it | Equals the average of the left and right sums |
The last row is the most useful fact here: the trapezoidal estimate is exactly the average of the left and right Riemann sums with the same subintervals. If a question gives you both, you never need to recompute anything.
The two error rules are genuinely different
A right Riemann sum overestimates an INCREASING function regardless of concavity. The trapezoidal rule overestimates a CONCAVE UP function regardless of whether it rises or falls. Quoting the wrong criterion loses the justification point even when the answer is right.
Frequently asked questions
Is the trapezoidal rule the average of the left and right sums?
Yes, exactly, when both use the same subintervals.
When does the trapezoidal rule overestimate?
When the function is concave up on the interval, because the chord lies above the curve.
Which is more accurate?
Usually the trapezoidal rule, since it uses two values per subinterval instead of one. The midpoint rule is often comparable.
In the CED: Unit 6: Integration and Accumulation