The Trapezoid Rule for Estimating Areas
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Question
State the trapezoid rule formula.
Answer
A ā (h/2)(yā + 2yā + 2yā + ... + 2yāāā + yā), where h = (b ā a)/n and yįµ¢ = f(a + iĀ·h).
š” 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 ā the horizontal width of each trapezoid strip. h = (b ā a) / n.
š” 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 ā it counts as a side of both.
š” Hint
Adjacent trapezoids share a boundary.
Question
Using the trapezoid rule with n = 2, estimate ā«[0 to 2] xĀ² dx.
Answer
h = 1. yā = 0, yā = 1, yā = 4. A ā (1/2)(0 + 2Ć1 + 4) = 0.5 Ć 6 = 3. (Exact = 8/3 ā 2.67)
š” Hint
x-values: 0, 1, 2. Find y = xĀ² 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.
š” 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.
š” 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 = (bāa)/n. 2. List all x-values: a, a+h, a+2h, ..., b. 3. Calculate yįµ¢ = f(xįµ¢) for each. 4. Apply: A ā (h/2)(yā + 2yā + ... + yā).
š” 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.
š” Hint
Trapezoids are exactly trapezoid-shaped ā they match straight lines perfectly.
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