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.
More to read
3.6 Numerical Integration
Some integrals refuse closed forms, so area is earned strip by strip. The trapezium rule joins sampled heights with straight tops and adds the resulting trapezia. Halving the width and recomputing shows the digits settling, and smooth functions settle fast. Four strips give a sketch; forty give a value to trust. Error behaviour guides the effort. For smooth curves the error shrinks like the square of the width, so each halving buys two extra binary digits of accuracy. Rough functions earn less per halving. Estimate first, halve once, and only refine while the answer keeps moving.
Approximating area: the Trapezium Rule
Sliders make over and under estimates visible. Drag the strip count up and watch the trapezium tops hug a convex curve from above, always slightly too generous. Flip to a concave curve and the same tops tuck underneath, always slightly stingy. The sketch predicts the sign of the error before any arithmetic. Use it to test halving claims. Record the estimate at 4 strips and at 8, then compare the errors against the exact value. The second error lands near a quarter of the first for smooth examples. Seeing the ratio once beats memorising the order claim.
Trapezoid rule
This companion demo isolates one strip at a time. Each trapezium area is average height times width, and the rule is just that idea summed with shared boundaries counted twice. Stepping through strip by strip shows where the doubling of interior ordinates comes from. The formula stops being a chant and becomes bookkeeping. Try a straight line input first: every strip is exact, so the total is exact. Then bend the curve and watch the small triangular gaps appear between chord and arc. Those gaps are the error, with their sign set by which way the curve bends.
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2 · Watch
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Where it sits
18 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.