Short on time? This page is the recap. For explanations, the figures and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- To estimate the area under a curve from a to b, split [a, b] into subintervals, find (width × height) on each, and add.
- Uniform partition: every width is Δx = (b − a)/n. Nonuniform: use each subinterval’s own width.
- The function can be given by a formula, a graph, a table or a description.
- With a table, a midpoint sum is possible only if the table gives values at the midpoints of the subintervals you use.
- In context, the sum estimates an accumulated change, with units (rate units) × (input units).
Key relationships
| Sum | One term on [xᵢ₋₁, xᵢ] |
|---|---|
| Left, L | f(xᵢ₋₁) × Δxᵢ |
| Right, R | f(xᵢ) × Δxᵢ |
| Midpoint, M | f(middle of the subinterval) × Δxᵢ |
| Trapezoidal, T | ½[f(xᵢ₋₁) + f(xᵢ)] × Δxᵢ |
Same partition: T = (L + R)/2.
| f on [a, b] | Too small | Too big |
|---|---|---|
| increasing | L | R |
| decreasing | R | L |
| concave up | M | T |
| concave down | T | M |
Assumptions behind the over/under rules
- The property (increasing, decreasing, concave up or down) must hold on the whole interval.
- A table alone does not show behaviour between readings; you need it stated, or information about f′ or f″.
- If f is linear, T and M are exact.
Mistakes to avoid
- Using Δx = (b − a)/n when the widths are unequal.
- Including the last value in a left sum, or the first value in a right sum.
- Averaging two table values and calling it a “midpoint” value.
- Using concavity to judge L or R, or monotonicity to judge M or T.
- Forgetting the ½ in trapezoid terms.
- Giving a value with no set-up on a calculator-active question.
Quick self-check
Let f(x) = x² on [1, 3] with two subintervals of width 1.
- Find L and R. (L = 1 + 4 = 5; R = 4 + 9 = 13)
- Find T and M. (T = 9; M = 1.5² + 2.5² = 8.5)
- The exact area is 26/3 ≈ 8.667. Which of the four sums are underestimates, and why? (L, because f is increasing on [1, 3]; and M, because f is concave up)
Next: practice questions.