The Trapezoid Rule for Estimating Areas
Practice Flashcards
Flip to reveal answersState the trapezoid rule formula.
Track your progress — Sign up free to save your progress and get smart review reminders based on spaced repetition.
All 8 Flashcards — The Trapezoid Rule for Estimating Areas
Sign up free to track progress and get spaced-repetition review schedules.
Question
State the trapezoid rule formula.
Answer
A ≈ (h/2)(y₀ + 2y₁ + 2y₂ + ... + 2yₙ₋₁ + yₙ), where h = (b − a)/n and yᵢ = f(a + i·h).
💡 Hint
Interior values are multiplied by 2. First and last by 1.
Question
What does h represent in the trapezoid rule?
Answer
h is the step width — the horizontal width of each trapezoid strip. h = (b − a) / n.
💡 Hint
b and a are the limits; n is the number of strips.
Question
Why do interior y-values get multiplied by 2 in the trapezoid rule?
Answer
Because each interior vertical line is shared by two adjacent trapezoids — it counts as a side of both.
💡 Hint
Adjacent trapezoids share a boundary.
Question
Using the trapezoid rule with n = 2, estimate ∫[0 to 2] x² dx.
Answer
h = 1. y₀ = 0, y₁ = 1, y₂ = 4. A ≈ (1/2)(0 + 2×1 + 4) = 0.5 × 6 = 3. (Exact = 8/3 ≈ 2.67)
💡 Hint
x-values: 0, 1, 2. Find y = x² at each.
Question
For a concave-up curve, does the trapezoid rule give an over- or underestimate?
Answer
Overestimate. The trapezoids sit above the curve, so the total estimated area is larger than the actual area.
💡 Hint
Think: concave up = smile = curve dips below the trapezoid.
Question
For a concave-down curve, does the trapezoid rule give an over- or underestimate?
Answer
Underestimate. The trapezoids fall below the curve, so the estimated area is smaller than the actual area.
💡 Hint
Think: concave down = frown = curve rises above the trapezoid.
Question
What are the 4 steps for applying the trapezoid rule?
Answer
1. Calculate h = (b−a)/n. 2. List all x-values: a, a+h, a+2h, ..., b. 3. Calculate yᵢ = f(xᵢ) for each. 4. Apply: A ≈ (h/2)(y₀ + 2y₁ + ... + yₙ).
💡 Hint
Write the y-values in a table to stay organised.
Question
When is the trapezoid rule exact (no error)?
Answer
When the function is linear (a straight line). Trapezoids perfectly fit straight-line sections with no gap or overlap.
💡 Hint
Trapezoids are exactly trapezoid-shaped — they match straight lines perfectly.
Read the notes
Full study notes for The Trapezoid Rule for Estimating Areas
Topic 5.8 hub
Trapezoid rule
More from Topic 5.8
All flashcards in this topic
Math AI exam skills
Paper structures & tips
Track your progress with spaced repetition
Sign up free — Aimnova tells you exactly which cards to review and when, so you remember everything before your IB exam.
Start Free