Simpson's Rule on a Cubic

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Simpson's Rule on a Cubic
Why is the Simpson's Rule approximation of a
degree 3 polynomial exact?
First, let's shift the function so that the
interval is centered at the origin.
The odd degree terms do not contribute a net
value to the area when the interval is about the
origin, so we can eliminate them.
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Thus the area under the curve is the same for
both functions shown.
This is not the quadratic used by Simpson's Rule,
as it does not pass through all of the right
points. It does match the cubic at x0, and is
off by the same amount, in opposite directions,
on both ends This difference can be reconciled by
the addition of a linear term, which again
contributes no net value to the area.
Simpson's quadratic