Compare integration rules with an exact displacement
Sampling and the weighting rule are both parts of a numerical integral.
Learning outcome: Compute one small grid by hand, compare all three methods and double N. Retain signed errors and state precisely why Simpson is exact for this supplied function.
Starting recap
Read derivatives as local rates and integrals as signed accumulation. Review power and chain rules, vector components, units and initial conditions where used.
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Try
With N=2, left gives3 m, trapezoid5 m and Simpson14/3 m. With N=4 the trapezoid error is one quarter of its N=2 error.
Open the experimentPredict
For this quadratic velocity, doubling N changes trapezoid error by a factor:
Experiment
Compute one small grid by hand, compare all three methods and double N. Retain signed errors and state precisely why Simpson is exact for this supplied function.
Integrate sampled velocity and compare error orders
Worked teaching model · synthetic values · Supplied smooth velocity v=1+t² m/s on0≤t≤2 s. Noise-free grid; compare quadrature with exact integral14/3 m. Simpson requires the registered even interval count. Exactness for this quadratic is not a universal zero-error claim.
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Velocity samples used by the quadrature rule
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: sampled smooth function; joining is a visual guide (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Velocity samples used by the quadrature rule
sampled smooth function; joining is a visual guide
| time (s) | velocity (m/s) |
|---|---|
| 0 | 1 |
| 0.5 | 1.25 |
| 1 | 2 |
| 1.5 | 3.25 |
| 2 | 5 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Approximate displacement | 4.75 | m |
| Exact displacement | 4.666667 | m |
| Signed integration error | 0.08333333 | m |
| Step size | 0.5 | s |
A sample count and a quadrature rule jointly determine the approximation.
The exact integral of1+t² from0 to2 is14/3. Left-rule error is−4/N+4/(3N²); trapezoid error is4/(3N²). Simpson integrates this quadratic exactly, not arbitrary functions or noisy observations. Doubling N asymptotically halves first-order left error and quarters trapezoid error; these are truncation claims, not calibration guarantees.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A sample count and a quadrature rule jointly determine the approximation.
The exact integral of1+t² from0 to2 is14/3. Left-rule error is−4/N+4/(3N²); trapezoid error is4/(3N²). Simpson integrates this quadratic exactly, not arbitrary functions or noisy observations. Doubling N asymptotically halves first-order left error and quarters trapezoid error; these are truncation claims, not calibration guarantees.
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Public practice checks a specific claim. It does not establish independent mastery.
How would a common velocity calibration offset affect every integration rule?
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