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) . ( . ) . ( . ) ( ) . ( ) ( ) ( ) . ( ) ( ) ( ) ( 76 3 23 5413 e 4 e 5 3 4 e 3 3 5 0 e 3 e 5 1 4 0 3 5 1 4 f 5 3 f 4 3 f 3 h 3 f 5 1 f 4 0 f 3 h dx x f dx x f I 8 7 6 6 3 2 1 4 3 3 0
Newton-Cotes Open Formula Newton-Cotes Open Formula Midpoint Rule Midpoint Rule ( ( One-point One-point ) ) ) ( ) ( ) ( ) ( ) ( ) ( ) ( f 24 a b 2 b a f a b x f a b dx x f 3 m b a a b x f(x) x m
Two-point Newton-Cotes Open Formula Two-point Newton-Cotes Open Formula Approximate by a straight line ) ( ) ( ) ( ) ( ) ( f 108 a b x f x f 2 a b dx x f 3 2 1 b a x 0 x 1 x f ( x ) x 2 h h x 3 h
Three-point Newton-Cotes Open Formula Three-point Newton-Cotes Open Formula Approximate by a parabola ) ( ) ( ) ( ) ( ) ( ) ( f 23040 a b 7 x f 2 x f x f 2 3 a b dx x f 5 3 2 1 b a x 0 x 1 x f ( x ) x 2 h h x 3 h h x 4
Newton-Cotes Open Newton-Cotes Open Integration Formulae Integration Formulae ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 6 7 5 4 3 2 1 4 5 4 3 2 1 4 5 3 2 1 3 2 1 3 1 f h 140 41 20 x f 11 x f 14 x f 26 x f 14 x f 11 a b 6 f h 144 95 24 x f 11 x f x f x f 11 a b 5 f h 45 14 3 x f 2 x f x f 2 a b 4 f h 4 3 2 x f x f a b 3 f h 3 1 x f a b 2 Error Truncation Formula n n a b h
Area under the function surface Double Integral Double Integral dy dx y x f dx dy y x f dy dx y x f d c b a b a d c d c b a      ) , ( ) , ( ) , (
T ( x, y ) = 2xy + 2x – x 2 – 2y 2 + 40 Two-segment trapezoidal rule Exact if using single-segment Simpson’s 1/3 rule (because the function is quadratic in x and y ) Double Integral Double Integral

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