The Trapezium Rule
Estimate a definite integral by slicing the region into trapezia of equal width, and use the curvature to say whether the estimate is above or below the true value.
What a learner can do afterwards
- Estimate an integral with four strips of equal width
- Halve the strip width and describe what happens to the error
- Decide from a sketch whether the estimate is too big or too small
1 · Read
Slice the region under the curve into n strips of equal width h and join the sampled heights with straight tops. Each strip is a trapezium, and adding them gives (h/2) times first plus twice each middle plus last. With n strips you need n + 1 height values, since neighbours share their boundaries.
For x squared from 0 to 4 with 4 strips, h = 1 and the heights are 0, 1, 4, 9, 16. The estimate is (1/2) times (0 + 2 times (1 + 4 + 9) + 16) = 22. The exact area is 64/3, about 21.33, so the estimate sits above it.
A sketch predicts the error sign before any arithmetic. Chords of a cup up convex curve sit above the arc, so the trapezia hold too much area and the estimate is too big. For a cap down concave curve the chords sag below and the estimate is too small. Halving the width roughly quarters the error for smooth curves, since the error scales with h squared.
For a straight line or a constant, every chord matches the graph, so the rule is exact. Root x bends cap down, so with 4 strips from 1 to 5 the estimate about 6.76 sits just under the exact 6.79. Only refine while the answer keeps moving.
Join the strip heights with chords, read the bend of the curve for the error sign, and halve the width to shrink the error.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.