Closed Newton-Cotes Formulas Thm.5.1.Assume thatx=x+kh are equally spaced nodes andx).The first four closed Newton-Cotes quadrature formulas are fe达=96+0 (trapezoidal rule) f达*6+4f+) (Simpson's rule) e*+3++) (Simpson's 3/8 rule) 1a=治%+2+12听+2+7 (Boole's rule)Closed Newton-Cotes Formulas 1 0 0 1 ( ) ( ) 2 x x h f x dx f f  +  2 0 0 1 2 ( ) ( 4 ) 3 x x h f x dx f f f  + +  3 0 0 1 2 3 3 ( ) ( 3 3 ) 8 x x h f x dx f f f f  + + +  Thm. 5.1. Assume that xk=x0+kh are equally spaced nodes and fk=f(xk ). The first four closed Newton-Cotes quadrature formulas are 4 0 0 1 2 3 4 2 ( ) (7 32 12 32 7 ) 45 x x h f x dx f f f f f  + + + +  (trapezoidal rule) (Simpson’s rule) (Simpson’s 3/8 rule) (Boole’s rule)