Key Idea: The trapezoid rule estimates the area under a curve using strips shaped like trapezoids. In the exam it usually comes together with the exact integral and a percentage error.
| Item | Rule |
|---|---|
| Strip width | h = (b − a)/n |
| Trapezoid rule | T = (h/2)(y₀ + 2y₁ +... + 2yₙ₋₁ + yₙ) |
| Exact area | ∫ f(x) dx, by hand or GDC |
| Percentage error | |approx − exact| / |exact| × 100% |
The usual question
- Estimate — Make a table of y-values at equally spaced x-values, then apply the rule.
- Exact — Write down the integral for the exact area and evaluate it.
- Error — Find the percentage error.
The two end heights count once; every middle height counts twice.
From x = 0 to x = 4 with 4 strips, what is h? 1.
The estimate is 5.8 and the exact area is 5.6. Percentage error? 0.2/5.6 × 100% ≈ 3.6%.
h′(x) = 0.6x² − 1.8x and h(4) = 1.4. Find h(x). h(x) = 0.2x³ − 0.9x² + 3.
Why is the trapezoid rule only an estimate? The straight tops of the strips do not follow the curve exactly.