Quadratures - Accuracy and Error Table

Essential Quadrature Rules

Rule	Points	degree of Polynomial $(N)$	Degree of Accuracy $(K)$	Order of Error/Consistency $(q)$
Left/Right Rectangle	1	0	$0$	$O(h)$
Midpoint	1	0	$1$	$O(h^2)$
Trapezoid	2	1	$1$	$O(h^2)$
Simpson/Kepler	3	2	$3$	$O(h^4)$
$K=0-1-1-3$, Order of Error $=1-2-2-4$

Degrees of Accuracy:

• Rectangle rules: K = 0 (only integrates constants exactly)
• Trapezoid & Midpoint: K = 1 (integrates up to linear functions exactly)
• Simpson/Kepler: K = 3 (integrates up to cubic functions exactly)

Composite Rules:

• Summed Rectangle: Error = O(h), halving h → error ÷ 2
• Summed Trapezoid: Error = O(h²), halving h → error ÷ 4
• Summed Simpson: Error = O(h⁴), halving h → error ÷ 16

Note: Rectangle rules are the simplest but least accurate. Midpoint rule is surprisingly good (same error order as trapezoid despite using only 1 point!).

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