Unit 6
Integration
17–20% AB & BCCED Unit 6
Riemann Sums
Left, right, midpoint, and trapezoidal approximations, plus over/underestimate rules.
Riemann Sum Setup
A Riemann sum approximates the area under from to . Divide into equal subintervals.
Partition Width
Sample Points
Left Riemann Sum
Left Endpoints
Use the left endpoint of each subinterval for the height.
Right Riemann Sum
Right Endpoints
Use the right endpoint of each subinterval for the height.
Midpoint Riemann Sum
Midpoints
Use the midpoint of each subinterval for the height.
Trapezoidal Sum
Trapezoids
Endpoints get coefficient 1, interior points get coefficient 2.
Limit Definition → Definite Integral
As n → ∞
As we take more and more rectangles, the Riemann sum approaches the exact area.
Over and Underestimates
Left/Right: determined by increasing/decreasing (first derivative).
Midpoint/Trapezoid: determined by concavity (second derivative).
Midpoint/Trapezoid: determined by concavity (second derivative).
| Function | Left | Right |
|---|---|---|
| Increasing | Under | Over |
| Decreasing | Over | Under |
| Concavity | Midpoint | Trapezoid |
|---|---|---|
| Concave Up | Under | Over |
| Concave Down | Over | Under |
⚠ Watch Out
Don't mix up the rules
Concavity does NOT matter for Left/Right. Increasing/decreasing does NOT matter for Midpoint/Trapezoid.AP® Calculus AB & BC · Unit 6 Overview · Mr. Brantley