5 · IB Maths AA
Trapezoidal rule and Riemann sums
Split the area under a curve into n trapezia or rectangles and watch how the approximation, the exact integral and the error change as n grows.
h = (b − a) / n, xᵢ = a + ih, yᵢ = f(xᵢ)
- Strip width h
- Approximation
- Exact F(b) − F(a)
- Error (approx − exact)
- Percentage error
- Verdict
Try this
- Choose x² on [0, 2] with the trapezoidal rule. Go from n = 4 to n = 8: by roughly what factor does the error shrink?
- Choose eˣ on [0, 2] with n = 4. Compare the trapezoidal and midpoint errors: how do their sizes and signs relate?
- Choose x³ − 3x on [−2, 2]. Why is the trapezoidal error zero for every n, even though the strips clearly miss the curve?
Exam tip: n strips need n + 1 ordinates, so list y₀ to yₙ in a table first; if the curve is concave up on the interval, the trapezoidal rule gives an overestimate.