Practice question · Multiple choice
Simpson's rule is exact for cubics although it fits only parabolas. Why does it get a free degree of accuracy?
Hints
- Integrate x³ over an interval symmetric about its midpoint. What do you get?
- Ask what an odd function contributes over a symmetric interval.
Show the answer
C. Because the cubic error term is odd about the panel's midpoint
Why
The leftover cubic term is odd about the midpoint, so its contributions on the two halves cancel exactly. That symmetry is why the error scales like h⁴ rather than h³, halving the step cuts the error sixteenfold, and the extra order came free from the geometry rather than from fitting anything cleverer.
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