AP Calculus AB and BC
Midpoint vs Trapezoidal Rule
The trapezoidal rule joins consecutive points with straight chords, so on a concave up curve it overestimates. The midpoint rule takes each rectangle's height from the centre of its subinterval and errs in the opposite direction, underestimating on a concave up curve.
Trapezoidal rule
Use when: You have data at listed points and no values in between, which is the usual table situation.
Midpoint rule
Use when: You can evaluate the function anywhere, so you can sample the centre of each subinterval.
Side by side
| Trapezoidal | Midpoint | |
|---|---|---|
| Shape used | Trapezoid | Rectangle at centre height |
| Concave up | Overestimate | Underestimate |
| Concave down | Underestimate | Overestimate |
| Works from a table | Yes | Only if midpoints are listed |
Both errors are governed by concavity, and they point opposite ways, which is why the true value is always trapped between them. On a concave up curve the chords sit above the graph while the midpoint rectangles cut under it.
The trapezoidal rule also equals the average of the left and right Riemann sums, which is a fast way to compute it when you already have both.
Unequal widths
The compact trapezoidal formula assumes equal spacing. With data at uneven intervals, compute each trapezoid separately as the average of its two heights times its own width.
Frequently asked questions
Which is more accurate?
For a smooth function the midpoint rule is usually about twice as accurate as the trapezoidal rule, but it requires being able to evaluate the function at the midpoints.
In the CED: Unit 6: Integration and Accumulation